Algebra Module 3 linear equations
Linear Inequality
contains a single variable on either side of the inequality symbol. x > 3 3x - 5 < 8 4x - 6 ≥ 8x + 12
Solve for y. x + 3y = 12 subtract x from both sides 3y = 12 - x divide both sides by 3 y = 4 - x/3 y = -x/3 + 4
y = -x/3 + 4
Solve for y. 4x - 12y = 84 subtract 4x from both sides -12y = 84 - 4x divide both sides by -12 y = 84/-12 + -4x/-12 y = -7 + 4/12x y = 1/3x - 7
y = 1/3x - 7
1/5 (3x + 4) = 2 (3 - 4x) + 12 3/5x + 4/5 = 6 - 8x + 12 combine like terms 3/5x + 4/5 = 18 - 8x add 8x to both sides 8 3/5x + 4/5 = 18 combine like terms 8 3/5x = 17 1/5 divide both sides by 8 3/5 x = 2
x = 2
Graph of an Inequality
is a picture that represents all of the solutions of the inequality. We graph inequalities by shading in all possible solutions to the inequality on a number line.
2x + 0 = 5 Ax + B = C
linear equation
2x + 3 = 1 Ax + B = C
linear equation
The sum of a number and fifty is one hundred eighty-eight.
n + 50 = 188
Solve for m. y = mx + d subtract d from both sides y-d = mx divide both sides by x y-d/x = m
y-d/x = m
Write in set-builder notation. The set of all digits greater than or equal to 2. Which of the following is a correct description of the given set given in set-builder notation? a. {0, 1, 2} b. {x | x is a digit greater than or equal to 2} c. {x | x is a digit greater than 2} d. {2, 3, 4, 5, 6, 7, 8, 9}
{x | x is a digit greater than or equal to 2}
x + 7 ≥ 3 subtract 7 from both sides x ≥ -4
(-4) ●----->
4 (3x + 4) > 2 (x + 2) 12x + 16 > 2x + 4 subtract 16 from both sides 12x > 2x - 12 subtract 2x from both sides 10x > -12 divide both sides by 10 x > -1 1/5
(-6/5) ○----->
0.2x - 0.4 (6 - x) = 0.3 0.2x - 2.4 + 0.4x = 0.3 combine like terms 0.6x - 2.4 = 0.3 add 2.4 to both sides 0.6x = 2.7 divide both sides by 0.6 x = 4.5
(0.2)(4.5) - 0.4 (6 - 4.5) = 0.3 0.9 - 2.4 + 1.8 = 0.3 -1.5 + 1.8 = 0.3 0.3 = 0.3
0.4x + 7.1 = 6.5 (10)(0.4x) + (10)(7.1) = (10)(6.5) 4x + 71 = 65 subtract 71 from both sides 4x = -6 divide both sides by 4 x = -1.5
(0.4)(-1.5) + 7.1 = 6.5 -0.6 + 7.1 = 6.5 (10)(-0.6) + (10)(7.1) = (10)(6.5) -6 + 71 = 65 65 = 65
0.6x - 1.3 = 4.1 (10)(0.6x) - (10)(1.3) = (10)(4.1) 6x - 13 = 41 add 13 to both sides 6x = 54 divide both sides by 6 x = 9
(0.6)(9) - 1.3 = 4.1 5.4 - 1.3 = 4.1 4.1 = 4.1
8x - 5 > 5 add 5 to both sides 8x > 10 divide both sides by 8 x > 1 1/4
(1 1/4) ○----->
1/3x + 1/4 = 7/12 LCD = 12 (12)(1/3x) + (12)(1/4) = (12)(7/12) 4x + 3 = 7 subtract 3 from both sides 4x = 4 divide both sides by 4 x = 1
(1/3)(1) + 1/4 = 7/12 1/3 + 1/4 = 7/12 LCD = 12 (12)(1/3) + (12)(1/4) = (12)(7/12) 4 + 3 = 7 7 = 7
9x = 3 (8x + 35) 9x = (3)(8x) + (3)(35) 9x = 24x + 105 subtract 24x from both sides -15x = 105 divide both sides by -15 x = -7
(9)(-7) = 3 ((8)(7) + 35) -63 = 3 (-56 + 35) -63 = -168 + 105 -63 = -63
If A = {multiples of 3} and B = {multiples of 4}, then A⊆B
False... A = {3, 6, 9, 12...} and B = {4, 8, 12, 16...}
x - 1 = -2 add -1 to both sides x = -1
-1 - 1 = -2 -2 = -2
-1 = -6 - 3x add 6 to both sides 5 = -3x divide both sides by -3 1 2/3 = x
-1 = -6 - 3(1 2/3) -1 = -6 - 5 -1 = -1
x - 2.9 = -12.9 add 2.9 to both sides x = -10
-10 - 2.9 = -12.9 -12.9 = -12.9
x/7 = -6 multiply both sides by 7 x = -42
-42/7 = -6 -6 = -6
0.7x = 0.63 divide both sides by 0.7 x = 0.9
0.7(0.9) = 0.63 0.63 = 0.63
To solve equation of the form Ax + B = C, when a, b, and c are real numbers, do the following:
1.) Get the variable term alone on one side of the equation. Use the Addition Property of Equality to add or subtract the same number from both sides. 2.) Get the variable alone on one side of the equation. Use the Multiplication Property of Equality to multiply or divide both sides of the equation by the coefficient of the variable. - if the coefficient is a fraction, multiply both sides by its reciprocal. 3.) Simplify, if needed, by combining like terms. 4.) Check your solution.
Thirteen minus four times a number is thirteen.
13 - 4n = 13
13 = 5x - 22 add 22 to both sides 35 = 5x divide both sides by 5 7 = x
13 = 5(7) - 22 13 = 35 - 22 13 = 13
14 = x - 7 add 7 to both sides 21 = x
14 = 21 - 7 14 = 14
15 + 2 = 3 + x + 6 combine like terms 17 = 9 + x subtract 9 from both sides 8 = x
15 + 2 = 3 + 8 + 6 17 = 17
15x - 7 = 2 + 12x subtract 2 from both sides 15x - 9 = 12x subtract 15x from both sides -9 = -3x divide both sides by -3 3 = x
15(3) - 7 = 2 + 12(3) 45 - 7 = 2 + 36 38 = 38
16 - 2x = -4x add 2x to each side 16 = -2x divide both sides by -2 -8 = x
16 + 2(-8) = -4(-8) 16 + 16 = 32 32 = 32
Sixteen minus seven-ninths of a number is twelve.
16 - 7/9n = 12
Solve. 18 - 8 + x = 8 + 5 - 3 combine like terms 10 + x = 13 - 3 10 + x = 10 subtract 10 from both sides x = 0
18 - 8 + 0 = 8 + 5 - 3 10 = 13 - 3 10 = 10
2 (x + 5) = -12 2(x) + 2(5) = -12 2x + 10 = -12 subtract 10 from both sides 2x = -22 divide both sides by 2 x = -11
2 (-11 + 5) = -12 2(-11) + 2(5) = -12 -22 + 10 = -12 -12 = -12
Two more than two times a number is five.
2 + 2n = 5
x - 1/6 = 11/6 add 1/6 to both sides x = 12/6 = 2
2 - 1/6 = 11/6 11/6 = 11/6
2x + 11 = 1 subtract 11 from both sides 2x = -10 divide both sides by 2 x = -5
2(-5) + 11 = 1 -10 + 11 = 1 1 = 1
2x + 3.6 = 6.2 subtract 3.6 from both sides 2x = 2.6 divide both sides by 2 x = 1.3
2(1.3) + 3.6 = 6.2 2.6 + 3.6 = 6.2 6.2 = 6.2
2x + 5x = 28 combine like terms 7x = 28 divide both sides by 7 x = 4
2(4) + 5(4) = 28 8 + 20 = 28 28 = 28
2/3 = 1/3 + 3x subtract 1/3 from both sides 1/3 = 3x divide both sides by 3 1/9 = x
2/3 = 1/3 + 3(1/9) 2/3 = 1/3 + 1/3 2/3 = 2/3
Which of the following can be solved using the Multiplication Property of Equality? a. x + 3 = 19 b. (x + y + z)(0.5) c. 3(x + 4) d. 2/3x = 5
2/3x = 5
Which of the following is an equation? a. 2x + y = 92 b. 5x - 4y c. 28x / 9 d. 2x
2x + y = 92
3.5 times a number is seventeen.
3.5n = 17
Thirty is five less than triple a number.
30 = 3n - 5
Triple a number is equal to eight more than five times the number.
3n = 8 + 5n
x/2 + 6 = 8 subtract 6 from both sides x/2 = 2 multiply both sides by 2/1 x = 4
4/2 + 6 = 8 2 + 6 = 8 8 = 8
6 + 4 + x = 2 + 8 combine like terms 10 + x = 10 subtract 10 from both sides x = 0
6 + 4 + x = 2 + 8 10 + 0 = 10 10 = 10
6x - 4 (6 - x) = 14 6x - 24 + 4x = 14 combine like terms 10x - 24 = 14 add 24 to both sides 10x = 38 divide both sides by 10 x = 3.8
6x - 4 (6 - x) = 14 22.8 - 4 (6 - 3.8) = 14 22.8 - 24 + 15.2 = 14 14 = 14
8 - 2 (x + 1) = 9 + x 8 - 2x - 2 = 9 + 1x combine like terms 6 - 2x = 9 + 1x subtract 1x from both sides 6 - 3x = 9 subtract 6 from both sides -3x = 3 divide both sides by -3 x = -1
8 - 2 (x + 1) = 9 + x 8 - 2 + 2 = 9 - 1 8 = 8
9x + 4 = 7x - 2 subtract 7x from both sides 2x + 4 = -2 subtract 4 from both sides 2x = -6 divide both sides by 2 x = -3
9(-3) + 4 = 7(-3) - 2 -27 + 4 = -21 - 2 -23 = -23
9x - 8x - 7 = 3 + x - 10 combine like terms 1x - 7 = x - 7 subtract 1x from both sides -7 = -7
9(-7) - 8(-7) - 7 = 3 + (-7) - 10 -63 + 56 - 7 = 3 + (-7) - 10 -7 - 7 = -4 - 10 -14 = -14
Which of the following would NOT be a step in solving an equation? a. Remove any parentheses by using the Distributive Property. b. Multiply of divide both sides of the equation to get the variable alone on one side of the equation. c. Add or subtract the number terms on both sides of the equation so that they are on the same side of the equation as the variable terms. d. Add or subtract terms on both sides of the equation to get all the variable terms on one side of the equation.
Add or subtract the number terms on both sides of the equation so that they are on the same side of the equation as the variable terms.
Solving Equations with Decimals Using the LCD
An equation containing decimals can be solved in a similar way. You can multiply both sides of the equation by an appropriate power of 10 to eliminate the decimal numbers and work only with integer coefficients.
Find the intersection and union of the sets A = {3, 5, 6, 8, 9} and B = {0, 1, 2, 4}
A⋂B = The solution set is Ø A⋃B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Examples of Formulas
C = 2πr... the formula for finding the circumference of a circle P = 2l + 2w... the formula for finding the perimeter I = Prt... the formula for finding simple interest
There are several ways to write a set:
C = {1, 2, 3, 4, 5, 6} C = {x | x is a natural number less than 7} C = {x | x ϵ N and x < 7} C = {x | x ϵ N and x ≥ 6}
Solve for r. C = 2πr divide both sides by 2π C/2π = r
C/2π = r
Which of the following statements about a formula is FALSE? a. A formula is an equation in which variables are used to describe a relationship. b. A formula is solved for a specified variable when that variable is alone on one side of the equation and the other side does not contain that variable. c. A formula can be solved for a specified variable in the same way that linear equations can be solved for a variable. d. Solving a formula for a specified variable required extra steps compared to solving linear equations for the variable.
Solving a formula for a specified variable required extra steps compared to solving linear equations for the variable.
Solving equations when simplifying is needed
Sometimes it is necessary to simplify one or both sides of the equation before getting the variable term on one side of the equation and the number term on the other side of the equation. Simplify by combining like terms that are on the same side of the equation.
What is the FIRST property of equality that should be used to simplify the equation 9x = 5x - 4 and how will it be used? a. The Multiplication Property of Equality should be used to divide each side of the equation by 5. b. The Multiplication Property of Equality should be used to divide each side of the equation by 9. c. The Addition Property of Equality should be used to add 5 to each side of the equation. d. The Addition Property of Equality should be used to subtract 5x from both sides of the equation.
The Addition Property of Equality should be used to subtract 5x from both sides of the equation.
Solving Equations with Fractions
The equation-solving procedures is the same for equations with or without fractions, however, takes care and can be time consuming.
Set Intersection
The intersection of two sets is the set of all the elements that are common to both sets. The intersection of A and B is written as A⋂B and includes those elements that are in set A and in set B. If A = {a, b, c, d, e} and B = {a, e, i, o, u}, then A⋂B = {a, e}
Least Common Denominator (LCD)
The least common denominator (LCD) of two or more fractions is the least common multiple (LCM) of the denominators of the fractions.
List all the elements for the set described. {x | x is an integer number between -3 and 5}
The solution set is {-2, -1, 0, 1, 2, 3, 4}
List all the elements for the set described. {x | x is a whole number less than 3}
The solution set is {0, 1, 2}
List all the elements for the set described. {x | x is a whole number less than 0}
The solution set is Ø
Determine if the following statement is true or false. {16, 17, 18, 19} ⊆ {15, 16, 17, 18, 19}
True
Determine whether the equation is a linear equation. -4x + 6 = 2
Yes
Solution
a solution of an equation is the number(s) that, when substituted for the variable(s), makes the equation true.
What is the FIRST goal when solving an equation with variables on both sides? a. Isolate the variable by performing the necessary operations to make the coefficient of the variable term to 1. b. Perform the necessary operations so that one side of the equation is equal to zero. c. If necessary, simplify by combining like terms on each side first, then rewrite the equation so that the variable terms are on one side of the equation. d. Rewrite the equation so that the number terms are on one side of the equation.
If necessary, simplify by combining like terms on each side first, then rewrite the equation so that the variable terms are on one side of the equation.
Remember, if the sign of the variable term is positive, us subtraction to reverse the operation.
If the sign of the variable term is negative, use addition to reverse the operation.
*NOTE
If you know the value of x, then the order of operation tells us to multiply before adding. When trying to solve for x, we must "undo" this. That is, we must add (or subtract) first, then multiply (or divide).
By following certain procedures, we can often transform an equation into a simpler equivalent equation that has the form of x = some number.
In this form, the number is a solution of the equation.
Just as an open circle is used on the number line to represent < or >, a parentheses is used in interval notation.
Likewise, a closed circle is used on the number line to represent ≤ or ≥, a bracket is used in interval notation.
What step would NOT be used in solving the equation 1/3 (6x + 3) = 1/4 (4x - 8)? a. Multiply of divide both sides of the equation to get the variable alone on one side of the equation. b. Remove any parentheses by using the Distributive Property. c. Add or subtract terms on both sides of the equation to get all the variable terms on one side of the equation. d. Multiply the least common denominator (LCD) after applying the Distributive Property.
Multiply the least common denominator (LCD) after applying the Distributive Property.
Determine whether the equation is a linear equation. 1/2x2 = 3x
No
Is -3 a solution for the equation 7x - 2 = 5 ? 7(-3) - 2 = 5 -21 - 2 = -23
No
Among the following steps for solving a formula for a specified variable, which would come FIRST? a. Multiply of divide both sides of the equation to get get the specified variable alone on one side of the equation. b. Remove any parentheses by using the Distributive Property. c. Add or subtract terms on both sides of the equation to get all terms not containing the specified variable on the other side of the equation. d. Add or subtract terms on both sides of the equation to get all terms containing the specified variable on one side of the equation.
Remove any parentheses by using the Distributive Property.
Choose the set that is NOT a subset of A = {10, 20, 30, 40, 50, 60, 70, 80}. a. P = (70} b. B = {10, 20, 30} c. S = {70, 80, 90} d. M = {10, 40, 80}
S = {70, 80, 90}
*NOTE
Addition and subtraction "undo" each other, meaning that adding and subtracting the same number result in no change.
x + 2 = 11 subtract 2 from both sides x = 9
9 + 2 = 11 11 = 11
Solve for Q. T = V/Q V/T = Q
V/T = Q
Solve for y. 7x + y = 7 subtract 7x from both sides y = -7x + 7
y = -7x + 7
11 - 26 = -5x combine like terms -15 = -5x divide both sides by -5 3 = x
11 - 26 = -5(3) -15 = -15
11.4 + x - 7.1 = 4.4 combine like terms 4.3 + x = 4.4 subtract 4.3 from both sides x = 0.1
11.4 + 0.1 - 7.1 = 4.4 4.4 = 4.4
Which equation shows the CORRECT way to simplify 3x-2/4 = 2/3x -1/4 using the LCD? a. 12 (3x - 2) = 12 (2x) - 12 (1) b. 12 (3x-2/4) = 12 (2/3x) - 12 (1/4) c. 3 (3x-2/4) = 3 (2/3x) - 4 (1/4) d. 4 (3x-2/4) = 4 (2/3x) - 4 (1/4)
12 (3x-2/4) = 12 (2/3x) - 12 (1/4)
The quotient of twelve and a number is one-ninth.
12 / n = 1/9
12 = 5x - 8 add 8 to both sides 20 = 5x divide both sides by 5 4 = x
12 = 5(4) - 8 12 = 20 - 8 12 = 12
By what number can 0.3x + 4.25 = 9.1 - 0.33x be multiplied to change the decimals to integers? a. 10 b. 100 c. 1,000 d. when solving an equation, there is no rule that allows you to eliminate decimals, only fractions
100
Solving equations with parentheses
For all real numbers a, b, and c, a(b + c) = ab + ac
Write the set in set-builder notation. G = {4, 6, 8, 10}
G = {x | x is even and 4 ≤ x ≤ 10}
1+4x/7 + 3-x/3 = 4/21 LCD = 21 (21)(1+4x/7) + (21)(3-x/3) = (21)(4/21) 3 (1 + 4x) + 7 (3 - x) = 4 3 + 12x = 21 - 7x = 4 combine like terms 24 + 5x = 4 subtract 24 from both sides 5x = -20 divide both sides by 5 x = -4
((1+(4)(-4))/7) + ((3-(-4)/3) = 4/21 (1+(-16)/7) + 7/3 = 4/21 -15/7 + 7/3 = 4/21 LCD = 21 (21)(-15/7) + (21)(7/3) = (21)(4/21) -45 + 49 = 4 4 = 4
0.50x + 0.15(30) = 37.5 0.50x + 4.5 = 37.5 (10)(0.50x) + (10)(4.5) = (10)(37.5) 5x + 45 = 375 subtract 45 from both sides 5x = 330 divide both sides by 5 x = 66
(0.50)(66) + 0.15(30) = 37.5 33 + 4.5 = 37.5 37.5 = 37.5
4x + 7/4 = 5x - 7/4 LCD = 20 (20)(4x) + (20)(7/4) = (20)(5x) - (20)(7/4) 80x + 35 = 100x - 35 subtract 80x from both sides 35 = 20x - 35 add 35 to both sides 70 = 20x divide both sides by 20 3.5 = x
(4)(3.5) + 7/4 = (5)(3.5) - 7/4 14 + 7/4 = 17.5 - 7/4 15.75 = 15.75
-x + 8 - x = 3x + 10 - 3 combine like terms -2x + 8 = 3x + 7 subtract 7 from both sides -2x + 1 = 3x add 2x to both sides 1 = 5x divide both sides by 5 0.2 = x
-1(0.2) + 8 - 1x(0.2) = 3(0.2) + 10 - 3 -0.2 + 8 -0.2 = 0.6 + 10 - 3 7.6 = 7.6
-1/2x + 10 = 16 subtract 10 from both sides -1/2x = 6 multiply both sides by -2/1 x = -12
-1/2(-12) + 10 = 16 6 + 10 = 16 16 = 16
-12x - 4x = -3 combine like terms -16x = -3 divide both sides by -16 x = 0.1875
-12(0.1875) - 4(0.1875) = -3 -2.25 - 0.75 = -3 -3 = -3
-14 = -x divide both sides by -1 14 = x
-14 = (14)(-1x) -14 = -14
-14 = x - 3 add 3 to both sides -11 = x
-14 = -11 - 3 -14 = -14
-15x = -195 divide both sides by -15 x = 13
-15(13) = -195 -195 = -195
In the list of numbers, find the one that is a solution of the given equation. 10, -16, 23 -3 = x + 13
-16
x - 1/2 = 5/2 add 1/2 to both sides x = -4/2 = -2
-2 - 1/2 = -5/2 -5/2 = -5/2
x + 19/31 = -4/31 subtract 19/31 from both sides x = -23/31
-23/31 + 19/31 = -4/31 -4/31 = -4/31
x/3 + 3 = x/5 - 1/3 LCD = 15 (15)(x/3) + (15)(3) = (15)(x/5) - (15)(1/3) 5x + 45 = 3x - 5 subtract 45 from both sides 5x = 3x - 50 subtract 3x from both sides 2x = -50 divide both sides by 2 x = -25
-25/3 + 3 = -25/5 - 1/3 -8 1/3 + 3 = -5 - 1/3 -5 1/3 = -5 1/3
-28x - 13 = 47 add 13 to both sides -28x = 60 divide both sides by -28 x = -2 1/7
-28(-2 1/7) - 13 = 47 60 - 13 = 47 47 = 47
-3 (x + 5) + 5 = 5 (x + 4) + 7 -3x - 15 + 5 = 5x + 20 + 7 combine like terms -3x - 10 - 5x + 27 add 10 to both sides -3x = 5x + 37 subtract 5x from both sides -8x = 37 divide both sides by -8 x = -4.625
-3 (x + 5) + 5 = 5 (x + 4) + 7 -3 (-4.625 + 5) + 5 = 5 (-4.625 + 4) + 7 13.875 -15 + 5 = -23.125 + 20 + 7 -1.125 + 5 = -3.125 + 7 3.875 = 3.875
-4x - 10 = 7x + 23 add 10 to both sides -4x = 7x + 33 subtract 7x from both sides -11x = 33 divide both sides by -11 x = -3
-4(-3) - 10 = 7(-3) + 23 12 - 10 = -21 + 23 2 = 2
x/3 = -15 divide both sides by 3 x = -45
-45/3 = -15 -15 = -15
In the list of numbers, find the one that is a solution of the given equation. 0, -5, -10 x + 5 = 0
-5
-5 (x - 3) + 7 = x - 8 (-5)(1x) + (-5)(-3) + 7 = x - 8 -5x + 15 + 7 = x - 8 combine like terms -5x + 22 = x - 8 add 8 to both sides -5x + 30 = x subtract 30 from both sides -5x = x - 30 subtract 1x from both sides -6x = -30 divide both sides by -6 x = 5
-5 (5 - 3) + 7 = 5 - 8 (-5)(5) + (-5)(-3) + 7 = 5 - 8 -25 + 15 + 7 = 5 - 8 -10 + 7 = 5 - 8 -3 = -3
-5.3 = 6.7 - 4x subtract 6.7 from both sides -12 = -4x divide both sides by -4 3 = x
-5.3 = 6.7 - 4(3) -5.3 = 6.7 - 12 -5.3 = -5.3
-5x + 4 = -4 (2x + 5) -5x + 4 = -8x - 20 add 8x to both sides 3x + 4 = -20 subtract 4 from both sides 3x = -24 divide both sides by 3 x = -8
-5x + 4 = -4 (2x + 5) (-5)(-8) + 4 = -4 ((2)(-8) + 5) 40 + 4 = -4 (-16 + 5) 44 = (-4)(-16) + (-4)(5) 44 = 64 - 20 44 = 44
-6 = x + 2 subtract 2 from both sides -8 = x
-6 = -8 + 2 -6 = -6
-6x = -32 + 2x subtract 2x from both sides -8x = -32 divide both sides by -8 x = 4
-6(4) = -32 + 2(4) 24 = -32 + 8 24 = 24
-8x + 7 = 31 subtract 7 from both sides -8x = 24 divide both sides by -8 x = -3
-8(-3) + 7 = 31 24 + 7 = 31 31 = 31
-8x = 40 divide both sides by -8 x = -5
-8(-5) = 40 40 = 40
-8x = 0 divide both sides by -8 x = 0
-8(0) = 0 0 = 0
-9/19 = x + 6/19 subtract 6/19 from both sides -15/19 = x
-9/19 = -15/19 + 6/19 -9/19 = -9/19
Which is NOT a step in solving -x + 2 = y for x? a. -x/-1 = y-2/-1 b. -x + 2 - y = y - y c. x = -y + 2 d. -x + 2 - 2 = y - 2
-x + 2 - y = y - y
0.05x - 3.05 = 0.4 (x - 5) 0.05x - 3.05 = (0.4)(1x) + (0.4)(-5) 0.05x - 3.05 = 0.4x - 2 add 3.05 to each side 0.05x = 0.4x + 1.05 subtract 0.4x from both sides -0.35x = 1.05 divide both sides by 0.35 x = -3
0.05x - 3.05 = 0.4 (x - 5) (0.05)(-3) - 3.05 = (0.4)(-3) + (0.4)(-5) -0.15 - 3.05 = -1.2 - 2 -3.2 = -3.2
0.2 (x - 8) = 3.6 0.2x - 1.6 = 3.6 add 1.6 to both sides 0.2x = 5.2 divide both sides by 0.2 x = 26
0.2 (x - 8) = 3.6 (0.2)(26) + (0.2)(-8) = 3.6 5.2 - 1.6 = 3.6 3.6 = 3.6
1 + 6 = x - 9 combine like terms 7 = x - 9 add 9 to both sides 16 = x
1 + 6 = 16 - 9 7 = 7
1 - 2x = -5x + 13 subtract 1 from both sides -2x = -5x + 12 add 5x to both sides 3x = 12 divide both sides by 3 x = 4
1 - 2(4) = -5(4) + 13 1 - 8 = -20 + 13 -7 = -7
Solve. 1 = x - 8 add 8 to both sides 9 = x
1 = 9 - 8 1 = 1
Solving an equation
1. Remove any parentheses by using the Distributive Property. 2. If fractions or decimals remain, multiply each term by the least common denominator (LCD) of all the fractions. 3. Simplify each side, if possible. 4. Add or subtract terms on both sides of the equation to get all the variable terms on one side of the equation. 5. Add or subtract number terms on both sides of the equation to get all the number terms on the other side of the equation. 6. Multiply or divide both sides of the equation to get the variable alone on one side of the equation. 7. Simplify the solution. 8. Check your solution.
Solving and Equation Using the Addition Property of Equality
1.) Add or subtract the same number from both sides of the equation to get the variable on one side of the equation by itself. - if a number is being added to x, use subtraction - if a number is being subtracted from x, use addition 2.) Simplify, if needed, by combining like terms. 3.) Check your solution.
Solving a Formula for a Specified Variable: To solve a formula or an equation for a specified variable, use the same steps that are used to solve an equation EXCEPT treat the specified variable as the only variable in the equation and treat the other variables as if they are numbers.
1.) Identify the variable you are solving for. 2.) Remove any parentheses by using the Distributive Property. 3.) If fractions or decimals remain, multiply each term by the least common denominator (LCD) of all the fractions. 4.) Simplify each side if possible. 5.) Add or subtract terms on both sides of the equation to get all terms containing the specified variable on one side of the equation. 6.) Add or subtract terms on both sides of the equation to get all the terms not containing the specified variable on the other side of the equation. 7.) Multiply or divide both sides of the equation to get the specified variable alone on one side of the equation. 8.) Simplify the solution.
To Graph a Linear Equality
1.) Plot the boundary point, which is the point that separates the solutions and the non-solutions. a.) If the boundary point is a solution (≥ or ≤), use a closed circle: <-----● ●-----> b.) If the boundary point is not a solution (> or <), use an open circle: <-----○ ○-----> 2.) Shade all numbers to the side of the boundary point that contains the solutions to the inequality.
Which of the following equations is equivalent to 1.2x - 5 + 3 = 3.2 - 5.6x + 0.4x immediately after combining like terms on both sides? a. 1.2x + 2 = 3.2 -6x b. 6.4x = 5.2 c. 1.2x - 2 = 5.6x + 0.4x d. 1.2x -2 = 3.2 - 5.2x
1.2x -2 = 3.2 - 5.2x
x+5/4 = x/2 + 9/10 LCD = 20 (20)(x+5/4) = (20)(x/2) + (20)(9/10) 5 (x + 5) = 10x + 18 5x + 25 = 10x + 18 subtract 10 from both sides -5x + 25 = 18 subtract 25 from both sides -5x = -7 divide both sides by -5 x = 1.4
1.4+5/4 = 1.4/2 + 9/10 6.4/4 = 1.4/2 + 9/10 LCD = 20 (20)(6.4/4) = (20)(1.4/2) + (20)(9/10) 32 = 14 + 18 32 = 32
1/4x - 2/3 = 5/12x LCD = 12 (12)(1/4x) + (12)(-2/3) + (12)(5/12x) 3x - 8 = 5x subtract 3x from both sides -8 = 2x divide both sides by 2 -4 = x
1/4x - 2/3 = 5/12x (1/4)(-4) -2/3 = (5/12)(-4) -1 2/3 = -1 2/3
1/5x + 8 = 12 subtract 8 from both sides 1/5x = 4 multiply both sides by 5/1 x = 20
1/5(20) + 8 = 12 4 + 8 = 12 12 = 12
1/8x + 3 = -1 subtract 3 from both sides 1/8x = -4 multiply both sides by 8/1 x = -32
1/8(-32) + 3 = -1 -4 + 3 = -1 -1 = -1
10 - 1/10x = 1/15x LCD = 30 (30)(10) - (30)(1/10x) = (30)(1/15x) 300 - 3x = 2x add 3x to both sides 300 = 5x divide both sides by 5 60 = x
10 - (1/10)(60) = (1/15)(60) 10 - 6 = 4 4 = 4
10x = -30 divide both sides by 10 x = -3
10(-3) = -30 -30 = -30
10x = -17 + 37 combine like terms 10x = 20 divide both sides by 10 x = 2
10(2) = -17 + 37 20 = 20
11 (x - 6) = -3 - 8 (11)(1x) + (11)(-6) = -3 - 8 11x - 66 = -11 add 66 to both sides 11x = 55 divide both sides by 11 x = 5
11 (5 - 6) = -3 - 8 55 - 66 = -3 - 8 -11 = -11
18 - 8 + x = 8 + 9 - 6 combine like terms 10 + x = 11 subtract 10 from both sides x = 1
18 - 8 + 1 = 8 + 9 - 6 10 + 1 = 11 11 = 11
2 = x + 9 subtract 9 from both sides -7 = x
2 = -7 + 9 2 = 2
2x + 18 = 3 + 5x - 6 combine like terms 2x + 18 = -3 + 5x add 3 to both sides 2x + 21 = 5x subtract 2x from both sides 21 = 3x divide both sides by 3 7 = x
2(7) + 18 = 3 + 5(7) - 6 14 + 18 = 3 + 35 - 6 32 = 32
x+5/7 = x/4 + 1/2 LCD = 28 (28)(x+5/7) = (28)(x/4) + (28)(1/2) 4 (x + 5) = 7x + 14 4x + 20 = 7x + 14 subtract 20 from both sides 4x = 7x - 6 subtract 7x from both sides -3x = -6 divide both sides by -3 x = 2
2+5/7 = 2/4 + 1/2 7/7 = 2/4 + 1/2 1 = 1
Which of the following is NOT part of the translation of "one-fourth of the difference of a number and seven is twenty-one" ? a. 1/4 b. = 21 c. (x - 7) d. 21x
21x
3 (0.5x - 4.2) = 0.6 (x - 12) (3)(0.5x) + (3)(-4.2) = (0.6)(1x) + (0.6)(-12) 1.5x - 12.6 = 0.6x - 7.2 add 12.6 to both sides 1.5x = 0.6x + 5.4 subtract 0.6x from both sides 0.9x = 5.4 divide both sides by 0.9 x = 6
3 ((0.5)(6) - 4.2) = 0.6 ((1)(6) - 12) 3 (3 - 4.2) = 0.6 (6 - 12) (3)(3) + (3)(-4.2) = (0.6)(6) + (0.6)(-12) 9 - 12.6 = 3.6 - 7.2 -3.6 = -3.6
3 (-x - 7) = -2 (2x + 5) 3(-1x) + 3(-7) = 2(2x) + 2(5) -3x - 21 = -4x - 10 add 21 to both sides -3x = -4x + 11 add 4x to both sides 1x = 11 x = 11
3 (-11 - 7) = -2 ((2)(11) + 5) -33 - 21 = -2 (22 + 5) -33 - 21 = -44 - 10 -54 = -54
x - 3/2 = 3/2 add 3/2 to both sides x = 3
3 - 3/2 = 3/2 3/2 = 3/2
Solve. 3x - 2 = -23 add 2 to both sides 3x = -21 divide both sides by 3 x = -7
3(-7) - 2 = -23 -21 - 2 = -23 -23 = -23
3x = 21 divide both sides by 3 x = 7
3(7) = 21 21 = 21
4 (x + 7) = 36 (4)(1x) + (4)(7) = 36 4x + 28 = 36 subtract 28 from both sides 4x = 8 divide both sides by 4 x = 2
4 (2 + 7) = 36 (4)(2) + (4)(7) = 36 8 + 28 = 36 36 = 36
4 (x - 7) + 12 = 2 (x - 6) (4)(1x) + (4)(-7) + 12 = (2)(1x) + (2)(-6) 4x - 28 + 12 = 2x -12 combine like terms 4x - 16 = 2x - 12 add 16 to both sides 4x = 2x + 4 subtract 2x from both sides 2x = 4 divide both sides by 2 x = 2
4 (2 - 7) + 12 = 2 (2 - 6) 8 - 28 + 12 = 4 - 12 -20 -+12 = 4 - 12 -8 = -8
4 (x + 1) = 28 4(1x) + 4(1) = 28 4x + 4 = 28 subtract 4 from both sides 4x = 24 divide both sides by 4 x = 6
4 (6 + 1) = 28 4(6) + 4(1) = 28 24 + 4 = 28 28 = 28
x + 16 = 20 subtract 16 from both sides x = 4
4 + 16 = 20 20 = 20
4 = -7 + 8x add 7 to both sides 11 = 8x divide both sides by 8 (11/8) or 1.375 = x
4 = -7 + 8(1.375) 4 = -7 + 11 4 = 4
4x - 9 = 2x + 19 add 9 to both sides 4x = 2x + 28 subtract 2x from both sides 2x = 28 divide both sides by 2 x = 14
4(14) - 9 = 2(14) + 19 56 - 9 = 28 + 19 47 = 47
4x + 15 = 5x + 13 subtract 15 from both sides 4x = 5x - 2 subtract 5x from both sides -1x = -2 divide both sides by -1 x = 2
4(2) + 15 = 5(2) + 13 8 + 15 = 10 + 13 23 = 23
4x + 6 = -6 + 2x + 18 combine like terms 4x + 6 = 12 + 2x subtract 6 for both sides 4x = 6 + 2x subtract 2x from both sides 2x = 6 divide both sides by 2 x = 3
4(3) + 6 = -6 + 2(3) + 18 12 + 6 = -6 + 6 + 18 18 = 18
4x - 39 = 3 - 3x subtract 3 from both sides 4x - 42 = -3x subtract 4x from both sides -42 = -7x divide both sides by -7 6 = x
4(6) - 39 = 3 - 3(6) 24 - 39 = 3 - 18 -15 = -15
Solve for x. y = 3/4x + 6 LCD = 4 (4)(y) = (4)(3/4x) + (4)(6) 4y = 3x + 24 subtract 24 from both sides 4y - 24 = 3x divide both sides by 3 4y/3 + -24/3 = x 4/3y - 8 = x
4/3y - 8 = x
Solve. 4/5 = 2/5 + 2x subtract 2/5 from both sides 2/5 = 2x divide both sides by 2 1/5 = x
4/5 = 2/5 + 2(1/5) 4/5 = 2/5 + 2/5 4/5 = 4/5
x/-8 = -5 multiply both sides by -8 x = 40
40/-8 = -5 -5 = -5
x/4 = 11 multiply both sides by 4 x = 44
44/4 = 11 11 = 11
Erica and Steven played a video game. Erica scored 8 less than 4 times Steven's score. Erica's score was 1,000 points. Let x = the number of points Steven scored. How many points did Steven score? a. 4x + 8 = 1,000 b. 1,000 - 4x = 8 c. 8 - 4x = 1,000 d. 4x - 8 = 1,000
4x - 8 = 1,000
5 (x + 1) - 3 (x - 3) = 17 5x + 5 - 3x + 9 = 17 combine like terms 2x + 14 = 17 subtract 14 from both sides 2x = 3 divide both sides by 2 x = 3/2 or 1 1/2
5 ((1 1/2) + 1) - 3 ((1 1/2) - 3) = 17 7 1/2 + 5 - 4 1/2 + 9 = 17 12 1/2 - 4 1/2 + 9 = 17 8 + 9 = 17 17 = 17
5 (2x - 1) + 2 = -2 (5)(2x) + (5)(-1) + 2 = -2 10x - 5 + 2 = -2 combine like terms 10x - 3 = -2 add 3 to both sides 10x = 1 divide both sides by 10 x = 1/10
5 ((2)(1/10) - 1) + 2 = -2 5 (1/5 - 1) + 2 = -2 1 - 5 + 2 = -2 -2 = -2
The sum of five and a number is thirty-nine.
5 + n = 39
5 = x + 13 subtract 13 from both sides -8 = x
5 = -8 + 13 5 = 5
The inequality x > 3 means that x could have the value of any number greater than 3.
5 > 3 = true statement, 5 is a solution to x > 3 0 > 3 = not a true statement, 0 is not a solution to x > 3
5x - 6 - 3x = 3x - 5 combine like terms 2x - 6 = 3x - 5 subtract 5 from both sides 2x - 1 = 3x subtract 2x from both sides -1 = x
5(-1) - 6 - 3(-1) = 3(-1) - 5 -5 - 6 + 3 = -3 - 5 -8 = -8
5x - 18x = 26 combine like terms -13x = 26 divide both sides by -13 x = -2
5(-2) - 18(-2) = 26 -10 + 36 = 26 26 = 26
5x + 26 - 6 = 9x + 12x combine like terms 5x + 20 = 21x subtract 5x from both sides 20 = 16x divide both sides by 16 1.25 = x
5(1.25) + 26 - 6 = 9(1.25) + 12(1.25) 6.25 + 26 - 6 = 11.25 + 15 26.25 = 26.25
5x + 3 = 18 subtract 3 from both sides 5x = 15 divide both sides by 5 x = 3
5(3) + 3 = 18 15 + 3 = 18 18 = 18
5x - 4 = 13 add 4 to both sides 5x = 17 divide both sides by 5 x = 3.4
5(3.4) - 4 = 13 17 - 4 = 13 13 = 13
56 - 9x = 11x add 9x to both sides 56 = 20x divide both sides by 20 2.8 = x
56 - 9(2.8) = 11(2.8) 56 - 25.2 = 30.8 30.8 = 30.8
Solve for h. r = 1/5hy divide both sides by y 5r/y = hy/y 5r/y = h
5r/y = h
What is the CORRECT application of the Distributive Property for 5 (x - 3) - 2 (x + 1) = 15? a. 5x - 3 - 2x + 1 = 15 b. 5x - 15 - 2x - 2 = 15 c. 5x - 15x - 2x - 2 = 15 d. 5x + 15 - 2x + 2 = 15
5x - 15 - 2x - 2 = 15
Choose the equation that is a linear equation in one variable. a. 5x3 + 3x = 4 b. 2 = 2 c. 5x = 2 d. 5xy + x = 3
5x = 2
To make the process of solving 1/2x - 4x = 1/3x easier, by what number can both sides of the equation be multiplied? a. 6 b. 2 c. 3 d. 4
6
6 (x - 5) + 4 = 4 (x - 4) 6x -30 + 4 = 4x - 16 combine like terms 6x - 26 = 4x - 16 subtract 4x from both sides 2x - 26 = +16 add 26 to both sides 2x - 10 divide both sides by 2 x = 5
6 (x - 5) + 4 = 4 (x - 4) 30 - 30 + 4 = 20 -16 4 = 4
6x - 8 = -2 add 8 to both sides 6x = 6 divide both sides by 6 x = 1
6(1) - 8 = -2 6 - 8 = -2 -2 = -2
6x - 7 = 23 add 7 to both sides 6x = 30 divide both sides by 6 x = 5
6(5) - 7 = 23 30 - 7 = 23 23 = 23
x + 2.7 = 9.4 subtract 2.7 from both sides x = 6.7
6.7 + 2.7 = 9.4 9.4 = 9.4
A motorcycle shop maintains an inventory of six times as many new bikes as used bikes. Currently there are 132 new bikes. Use n to represent the unknown number of used bikes.
6n = 132
x-7/2 = 1 - x/7 LCD = 14 (14)(x-7/2) = (14)(1) - (14)(x/7) (7)(x-7/1) = (14)(1) - (2)(x/1) 7 (x - 7) = 14 - 2x 7x - 49 = 14 - 2x add 2x to both sides 9x - 49 = 14 add 49 to both sides 9x = 63 divide both sides by 9 x = 7
7-7/2 = 1 - 7/7 LCD = 14 (14)(7-7/2) = (14)(1) - (14)(7/7) 14(0) = 14 - 14 0 = 0
8 - 7x - 2 = -4 + 5x - 14 combine like terms -7x + 6 = -18 + 5x subtract 6 from both sides -7x = -24 + 5x subtract 5x from both sides -12x = -24 divide both sides by -12 x = 2
8 - 7(2) - 2 = -4 + 5(2) - 14 8 - 14 - 2 = -4 + 10 - 14 -8 = -8
8x - 4 = -36 add 4 to both sides 8x = -32 divide both sides by 8 x = -4
8(-4) - 4 = -36 -32 - 7 = -36 -36 = -36
8/5x - 6/7 = 7x+1/2 LCD = 70 (70)(8/5x) - (70)(6/7) = (70)(7x+1/2) 112x - 60 = 35 (7x + 1) 112x - 60 = 245x + 35 add 60 to both sides 112x = 245x + 95 subtract 245x from both sides -133x = 95 divide both sides by -133 x = -5/7
8/5 (-5/7) - 6/7 = ((7)(-5/7) + 1)/2 -1 1/7 - 6/7 = -4/2 -1 1/7 - 6/7 = -2 -2 = -2
Solve for x. y = 5/8x - 5 LCD = 8 (8)(y) = (8)(5/8x) + (8)(-5) 8y = 5x - 40 add 40 to both sides 8y + 40 = 5x divide both sides by 5 8y/5 + 40/5 = x 8/5y + 8 = x
8/5y + 8 = x
9x = 8 + 7x subtract 7x from both sides 2x = 8 divide both sides by 2 x = 4
9(4) = 8 + 7(4) 36 = 8 + 14 36 = 36
9x = 6x + 15 subtract 6x from both sides 3x = 15 divide both sides by 3 x = 5
9(5) = 6(5) + 15 45 = 30 + 15 45 = 45
An inequality is a statement that shows the relationship between any two real numbers that are not equal. "NOTE: Inequalities can also be used to express the relationship between a variable and a number.
< "is less than" > "is greater than" ≤ "is less than or equal to" ≥ "is greater than or equal to"
4x - 7 > 9x + 3 add 7 to both sides 4x > 9x + 10 subtract 9x from both sides -5x > 10 divide both sides by -5 change the direction of the inequality x < -2
<-----○ (-2)
5 - 3x > 11 subtract 5 from both sides -3x > 6 divide both sides by -3 change the direction of the inequality x < -2
<-----○ (-2)
Graph the inequality on a number line. x < -6 Choose the correct answer. a. (-6) ○-----> b. <-----● (-6) c. (-6) ●-----> d. <-----○ (-6)
<-----○ (-6)
5x + 2 < 12 subtract 2 from both sides 5x < 10 divide both sides by 5 x < 2
<-----○ (2)
-2 + 2x < -x + 2 add 2 to both sides 2x < -1x + 4 add 1x to both sides 3x < 4 divide both sides by 3 x < 4/3
<-----○ (4/3)
3x - 1 ≤ -13 add 1 to both sides 3x ≤ -12 divide both sides by 3 x ≤ -4
<-----● (-4)
Graph the inequality on a number line. x ≤ -5/2 Choose the correct answer. a. (-5/2) ●-----> b. <-----● (-5/2) c. (-5/2) ○-----> d. <-----○ (-5/2)
<-----● (-5/2)
x ≤ -9
<-----● (-9)
5 - 4x ≥ -7 subtract 5 from both sides -4x ≥ -12 divide both sides by -4 change the direction of the inequality x ≤ 3
<-----● (3)
"Set A is the set of all natural numbers greater than 4"
A = {x | x is a natural number greater than 4} A = {x | x ϵ N and x > 4}
A = {a, e, i, o, u}
A = {x | x is a vowel in the alphabet}
Which of the following statements about linear equations in one variable is CORRECT? a. The numbers in a linear equation must be whole numbers. b. A linear equation has a variable with an exponent greater than 1. c. None of the numbers in a linear equation can ever be zero. d. A linear equation in one variable can be written in the term Ax + B = C.
A linear equation in one variable can be written in the term Ax + B = C.
Set
A set is a collection of like objects called elements. The symbol ϵ means "is an element of the set." For example, Ringo ϵ Beatles.
Infinity - an infinite set is a set whose elements cannot be counted.
An example of this is the set of real numbers. The set of real numbers increases without bound to the right and decreases without bound to the left on a number line.
Find the intersection and union of the sets A = {2, 6, 7, 8, 9} and B = {6, 7, 8, 9}
A⋂B = {6, 7, 8, 9} A⋃B = {2, 6, 7, 8, 9}
Which of the following shows the union of A = {1, 2, 3, 5, 7, 9, 10, 11} and B = {2, 3, 4, 6, 8, 10, 11}? a. A⋃B = {2, 3, 10, 11} b. A⋃B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} c. A⋃B = {2, 3, 4, 10, 11} d. A⋃B = {1, 2, 3, 5, 7, 11}
A⋃B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
When the equation 2x - x + 1 = x + 3 - 2 is solved, the result is 1 = 1. Which of the following is true? Select all that apply. a. The equation is an identity. b. The equation cannot be solved. c. There are an infinite number of solutions to the equation. d. There is no solution to the equation.
Both a and c, the equation is an identity and there are an infinite number of solutions to the equation.
Find the intersection and union of the sets B = {2, 3, 6, 9} and C = {1, 4, 5, 7, 8}
B⋂C = The solution is Ø B⋃C = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Write the set in set-builder notation. C = {Africa, Antarctica, Asia, Australia} a. C = {x | x is a continent} b. C = {x | x is a continent and x ends with America} c. C = {x | x is a continent and x begins with A} d. C = {x | x is a continent and x does not begin with A}
C = {x | x is a continent and x begins with A}
When solving an equation, simplify both sides of the equation whenever possible.
Combining like terms on both sides of the equation will make it easier to work with.
1.) Understand the problem. 2.) Choose a variable to represent the unknown quantity. 3.) Write an expression to represent each unknown quantity in terms of the variable. Look for key words to help you translate the words into algebraic symbols and expressions. 4.) Use a given relationship in the problem or an appropriate formula to write an equation. 5.) Write the equation.
EXAMPLE: One-third of a number is fourteen. 1/3 x n = 14 1/3n = 14 Five more than six times a number is three hundred five. 5 + 6 x n = 305 5 + 6n = 305 The larger of two numbers is three more than twice the smaller number. The sum of the numbers is thirty-nine. Larger number = 3 + 2s s + 3 + 2s = 39
Examine what occurs when you add, subtract, multiply, or divide both sides of the inequality by a negative number.
EXAMPLE: 4 < 6 Add -2 to both sides = 2 < 4 Subtract -2 from both sides = 6 < 8 Multiply both sides by -2 = -8 > -12 Divide both sides by -2 = -2 > -3 Whenever both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality is reversed.
How do you solve the inequality 3x + 7 ≥ 13? First, you must examine what occurs when you add, subtract, multiply, or divide both sides of the inequality by a positive number.
EXAMPLE: 4 < 6 Add 2 to both sides = 6 < 8 Subtract 2 from both sides = 2 < 4 Multiply both sides by 2 = 8 < 12 Divide both sides by 2 = 2 < 3 Each of the resulting inequalities are TRUE statements.
Determine if the following statement is true or false. 8 ϵ {x | x is an odd number}
False
Which of the following is the definition of the Distributive Property? a. For all real numbers a, b, and c, a (b + c) = ab + ac b. For all real numbers a, b, and c, if a = b, then a + c = b + c c. For all real numbers a, b, and c with c ≠ 0, if a = b, then ca = cb d. For all real numbers a, b, and c with c ≠ 0, if a = b, then a/c = b/c
For all real numbers a, b, and c, a (b + c) = ab + ac
The equation 3 (2x - 1) = 6 has been simplified to 6x - 3 = 6. What is the next step in solving the equation? a. Get the number terms on one side of the equation. b. Divide each side of the equation by the number in front of the variable. c. Get the variable term on one side of the equation. d. Simplify each side.
Get the number terms on one side of the equation.
Alisa solved an inequality and found x > 5. This is the graph of her solution. Is it correct or incorrect, and why? (5) ○-----> a. Her graph is incorrect because the boundary point is open and the graph does not shade values less than 5. b. Her graph is incorrect because the graph does not shade values less than 5. c. Her graph is correct because the boundary point is open and the graph shades values greater than 5. d. Her graph is incorrect because the boundary point is open
Her graph is correct because the boundary point is open and the graph shades values greater than 5.
Which formula can be solved using only the Multiplication Property of Equality? a. -8x + 7y = z, for x b. 2 (a + b - c) = 3, for a c. I = Prt, for P d. P = 2L + 2W, for W
I = Prt, for P
Which of the following represents the Addition Property of Equality? a. x - a = b b. a(b + c) = ab + ac c. If a = b, then a + b = b + c d. x + a = b
If a = b, then a + b = b + c
The Multiplication Property of Equality
If both sides of an equation are multiplied by the same non-zero number, the solution does not change. a, b, and c with c ≠ 0, if a = b, then ca = cb
Which of the following does NOT represent a step for solving equations using both the Addition and Multiplication Properties of Equality together? a. If the coefficient of the variable term is a fraction, simplify by subtracting the fraction from each side. b. Get the variable alone on one side of the equation by either multiplying both sides of the equation by the reciprocal of the coefficient of the variable (if it is a fraction) or dividing both sides of the equation by the coefficient of the variable. c. Get the variable term alone on one side of the equation by by using the Addition Property of Equality. d. Check the solution by substituting the resulting value for the variable into the original equation.
If the coefficient of the variable term is a fraction, simplify by subtracting the fraction from each side.
*NOTE
If the decimals are tenths, multiply by 10; if the decimals are hundredths, multiply by 100, etc...
The Addition Property of Equality
If the same number is added to both sides of an equation, the results on both sides are equal in value. That is, adding the same number to both sides of an equation, does not change the solution. If a = b, then a + c = b + c
Solving Equations in the Form ax + b = cx + d
In some cases, a term with a variable may appear on both sides of the equation. In these cases, it is necessary first to rewrite the equation so that all the terms containing the variable appear on one side of the equation. We do this by adding or subtracting one of the variable terms from both sides.
Find the intersection and union of the sets C and D C = {odd numbers between 1 and 11, inclusive} C = {1, 3, 5, 7, 9, 11} D = {Multiples of 3 between 1 and 15, inclusive} D = {3, 6, 9, 12, 15}
Intersection = C⋂D = {3, 9} Union = C⋃D = {1, 3, 5, 6, 7, 9, 11, 12, 15}
What is an equation? a. It is any number, variable, or product of numbers and/or variables. b. It is a combination of numbers, variables, operation symbols, and grouping symbols. It does not include an equal sign. c. It is a letter or symbol that represents an unknown quantity. d. It is a mathematical statement that two expressions are equal. It always contains an equal sign.
It is a mathematical statement that two expressions are equal. It always contains an equal sign.
Interval Notation is another way to represent the solution to an inequality.
It uses two values -- the starting point and the end point of the interval representing the solution. These two values are written inside (parentheses) and/or [brackets].
Which of the following is NOT a step in solving 2/3 (3x - 1) = 9/2 ? a. Multiply each side of the equation by the LCD. b. Use the Distributive Property to simplify. c. Use the Addition Property of Equality. d. Multiply each fraction by its reciprocal.
Multiply each fraction by its reciprocal.
Which is the SECOND step in solving 1/2x + 6 = -4 ? a. Add 4 to each side of the equation b. Multiply each side of the equation by 2 c. Divide each side of the equation by -4 d. Subtract -6 from each side of the equation
Multiply each side of the equation by 2
Which step shows how to solve 1/3x = 9? a. Multiply each side of the equation by 3 b. Subtract 1/3 from each side of the equation c. Divide each side of the equation by 9 d. Multiply each side of the equation by 1/3
Multiply each side of the equation by 3
Natural Numbers
Natural numbers are all whole numbers, excluding zero. The set of natural numbers is usually represented by N. The set of natural numbers could be written as N = {1, 2, 3, 4, 5...} Sets are represented by capital letters. The elements in the set are contained in braces and separated by commas. The set of natural numbers in an infinite set. The list of numbers continues forever, and is indicated by three dots, which are also known as an ellipsis.
For the equation 7x + 6 = 27, which of the following statements is FALSE? a. The Addition Property of Equality is needed to solve the equation. b. The Multiplication Property of Equality is needed to solve the equation. c. The solution can be checked by substituting the final value back into the original equation. d. Neither the Addition Property of Equality nor the Multiplication Property of Equality is needed to solve the equation.
Neither the Addition Property of Equality nor the Multiplication Property of Equality is needed to solve the equation.
Is -8 a solution of x + 12 = 20 ?
No -8 + 12 = 4 4 ≠ 20
Choose the infinite set. a. C = {x | x is an integer between -3 and 5} b. O = {1, 3, 5, 7, 9, 11...} c. C = {x | x is a member of US Congress} d. A = {1, 2, 3, 4, 5, 6, 7, 8}
O = {1, 3, 5, 7, 9, 11...}
Solve for m. P = m + n + k subtract k from both sides P - k = m + n subtract n from both sides P - k - n = m
P - k - n = m
Solve for C. P = 2D + 2C subtract 2D from both sides P - 2D = 2C divide both sides by 2 P-2D/2 = C
P-2D/2 = C
A = {consonants} B = {letters of the alphabet}
Set A is a subset of B. this can be written as A⊆B.
Set B is the set of natural numbers between -6 and 0.
Set B is an empty set, or a null set. We can write this as { } or Ø.
*NOTE
Since subtraction can be defined in terms of addition, this property works for subtraction as well as addition. If a = b, then a - c = b - c
Which of the following is NOT part of the procedure for writing an equation from a word problem? a. Solve for the unknown quantity b. Write an expression to represent each unknown quantity in terms of the variable c. Choose a variable to represent each unknown quantity d. Use a given relationship or an appropriate formula to write an equation
Solve for the unknown quantity
Examples of Solving Formulas: Solve for t. d = rt divide both sides by r d/r = t Solve for r. C = 2πr divide both sides by 2π C/2π = r Solve for v. a = v/t multiply t on both sides (t)a = (t)v/t ta = v
Solve for x. x + y = 8 subtract y from both sides x = 8 - y Solve for a. a - b = -3 add b to both sides x = -3 + b Solve for y. 5x + 3y = 6 subtract 5x from both sides 3y = 6 - 5x divide both sides by 3 y = 6-5x/3 Solve for x. y = mx + b subtract b from both sides y - b = mx divide both sides by m y-b/m = x
When 5x + 1 < 11 is solved, the solution is x < 2. Describe the graph of the solution. a. The boundary point is a closed circle at 2, and the graph shades values less than 2. b. The boundary point is a closed circle at 2, and the graph shades values greater than 2. c. The boundary point is an open circle at 2, and the graph shades values greater than 2. d. The boundary point is an open circle at 2, and the graph shades values less than 2.
The boundary point is an open circle at 2, and the graph shades values less than 2.
When writing the notation for an empty set, only use one of the symbols -- either { } or Ø, but not both.
The notation { Ø } means "a set whose only element is the empty set."
Solving the Equation
The process of finding the solution(s) of an equation is called solving the equation. The goal of solving the equation is to get the variable alone on one side of the equation. x = some number or some number = x
Set Union
The union of two sets is the set of every element that is in either or both sets. The union of sets A and B is written as A⋃B and includes those elements in either set A, set B, or both. If A = {a, b, c, d, e} and B = {a, e, i, o, u}, then A⋃B = {a, b, c, d, e, i, o, u}
-7 (3x - 9) + 2 = 21x + 65 -21x + 63 + 2 = 21x + 65 combine like terms -21x + 65 = 21x + 65 subtract 65 from both sides -21x - 21x
There is an infinite number of solutions to this equation
1/8 (x + 8) = 1/16 (2x + 16) 1/8x + 1 = 1/8x + 1
There is an infinite number of solutions to this equation
5 (4x - 20) = 20 (x - 5) 20x - 100 = 20x - 100
There is an infinite number of solutions to this equation
7 (x + 3) - 8 = 19 - 2x - 6 + 9x combine like terms 7x + 21 - 8 = 19 - 2x - 6 + 9x combine like terms 7x + 13 = 13 + 7x
There is an infinite number of solutions to this equation
When the equation 2x - 6 = 2x + 9 is solved, the result is -6 = 9. Which of the following is true? a. The equation is an identity. b. There is no solution to the equation. c. There are two solutions to the equation. d. There are an infinite number of solutions to the equation.
There is no solution to the equation.
-6 + 6 (x - 2) = 14x + 4 - 8x combine like terms -6 + 6x - 12 = 14x + 4 - 8x combine like terms -18 + 6x = 6x + 4 add 18 to both sides 6x = 6x + 22 subtract 6x from both sides 0 = 22
There is no solution to this equation
8 (4x - 6) = 4 (8x + 5) 32x - 48 = 32x + 20 add 48 to both sides 32x = 32x + 68 subtract 32x from both sides 0 = 68
There is no solution to this equation
8x + 9 = 8 (x + 5) 8x + 9 = 8x + 40 subtract 8x from both sides 9 = 40
There is no solution to this equation
x/7 - 3 = x/7 LCD = 21 (21)(x/7) + (21)(-3) + (21)(x/7) 3x - 63 = 3x subtract 3x from both sides 0 = -63
There is no solution to this equation
Negative Infinity - the -∞ symbol is used to denote negative infinity.
This is used when the set continues without end in the negative direction on a number line. Example: (-∞, a]
Positive Infinity - the ∞ symbol is used to denote positive infinity.
This is used when the set continues without end in the positive direction on a number line. Example: [ a, ∞)
Another way to write a set is using set-builder notation.
This is useful when the individual elements are not easily written.
To make the calculations a little easier, we can perform an extra step that will allow us to rewrite the given equation with fractions as an equivalent equation that does not contain fractions.
To make the process of solving equations with fractions easier, multiply both sides of the equation by the least common denominator (LCD) of all the fractions contained in the equation. Then use the Distributive Property to multiply each term in the equation by the LCD. If done correctly, all fractions will change into integers.
Determine whether the following statement is true or false. If B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6, 7}, then B⊆C.
True
If A = {odd numbers} and B = {integers}, then A⊆B
True... A = {1, 3, 5, 7...} and B = {...-2, -1, 0, 1, 2, 3, ...}
Intersections and Unions
Two operations that are used with sets are the intersection and the union.
Solving an Inequality
Use the same procedure to solve an inequality that is used to solve an equation, EXCEPT the direction of an inequality must be reversed if you multiply or divide both sides of the inequality by a negative number.
Which answer CORRECTLY finishes the statement. In order to solve an equation with parentheses, the first step is to simplify by _____. a. Getting the number terms on one side of the equation. b. Using the Distributive Property to remove the parentheses. c. Getting the variable term on one side of the equation. d. Combining like terms on each side of the equation.
Using the Distributive Property to remove the parentheses.
Why is 0 not a solution to the equation 2x + 1 = 5 ? a. The equation is 2x + 1 = 5, not 2x + 1 = 0. As a result, 0 is not a solution. b. The equation has no solution. c. When 0 is substituted for the variable, the equation is not true. d. 0 can never be the solution to an equation.
When 0 is substituted for the variable, the equation is not true.
Subsets
When all of the elements of one set are contained in another set, the smaller set is a subset of the larger set.
Solving an Inequality
When we solve an inequality, we are finding all the values that make the inequality true.
Which of the following statements is FALSE? a. The symbol ⊆ means "is a subset of." b. An infinite set contains an ellipsis to indicate the list goes on forever. c.When writing the notation for an empty set, use { Ø }. d. A set is a collection of like objects called elements.
When writing the notation for an empty set, use { Ø }.
Alexis solved the equation 8x - 2 = 4x - 1 and got the solution x = 0. How does she know this is NOT the correct answer? a. The equation cannot be multiplied or divided by the number zero. b. She needs two solutions to the equation because she has variables on both sides. c. Zero can never be the solution to an equation. d. When zero is substituted into the equation, the equation is not true.
When zero is substituted into the equation, the equation is not true.
Set X is the set of all natural numbers between 4 and 10. Set Y is the set of all natural numbers between 2 and 7, inclusive.
X = {5, 6, 7, 8, 9} Y = {2, 3, 4, 5, 6, 7}
If the elements in a set can be counted, the set is called a finite set. Otherwise, the set is infinite.
Y = {2, 3, 4, 5, 6, 7} is a finite set N = {1, 2, 3, 4, 5...} is an infinite set
Determine whether the equation is a linear equation. x/9 - 5 = 3
Yes
Is 2 a solution of the equation 3x - 1 = 5 ? Substitute 2 for x 3(2) - 1 = 5 6 - 1 = 5 5 = 5
Yes
Is 1/3 a solution of (x + 2)/(3 - x) = 7/8 ?
Yes (2 1/3)/(2 2/3) = 7/8
Is -11 a solution of -8x = 77 - x ?
Yes -8(-11) = 77 - (-11) 88 = 88
Solving equations in the form Ax + B = C
You must use both the Addition Property of Equality and the Multiplication Property of Equality together.
Equations with an Infinite Number of Solutions
an equation has an infinite number of solutions if the equation is always true, no matter the value of x. The solution of such equations is all real numbers.
Equations with No Solutions
an equation has no solution if there is no value of x that makes the equation true. The symbol used to show no solution is Ø.
Solve 5x + 2 = 17 and ax + b = c for x 5x + 2 = 17 subtract 2 from each side 5x = 15 divide both sides by 5 x = 3
ax + b = c subtract b from both sides ax = c - b divide both sides by a x = c-b/a
Equivalent Equation
equations that have exactly the same solutions.
Variable
is a letter or symbol that represents an unknown quantity.
Equation
is a mathematical statement that two expressions are equal. All equations contain an equal sign ( = )
Formula
is an equation in which variables are used to describe a relationship.
Linear Equation
is an equation that can be written in the form Ax + B = C, where A, B, and C are real numbers and A ≠ 0
Contradiction
is an equation that is false for all values. That is, when different values appear on both sides of the equal sign, we call the equation a contradiction.
Identity
is an equation that is true for all values. That is, when the same values appear on both sides of the equal sign, we call the equation an identity.
Solution of an Inequality
is any number that makes the inequality true.
The difference between a number and two is twenty-one.
n - 2 = 21
A number decreased by six is seventeen.
n - 6 = 17
Five equals the quotient of a number and seven.
n / 7 = 5
The quotient of a number and negative eight, increased by three, is twenty-seven.
x / (-8) + 3 = 27
1/3 (x - 3) = 4x - 3 (x - 1) + 4/3 1/3x - 1 = 4x - 3x + 3 + 4/3 combine like terms 1/3x - 1 = 1x + 4 1/3 add 1 to both sides 1/3x = 1x + 5 1/3 subtract 1x from both sides -2/3x = 5 1/3 divide both sides by -2/3 x = -8
x = -8
7x + 5 = 2 (3x - 2) + 9 7x + 5 = 15x - 4 + 9 combine like terms 7x + 5 = 15x + 5 subtract 5 from both sides 7x = 15x subtract 15x from both sides -8x = 0 divide both sides by -8 x = 0
x = 0
-1/6 (x - 12) + 1/3 (x + 3) = x - 7 -1/6x + 2 + 1/3x + 1 = x - 7 combine like terms 1/6x + 3 = 1x - 7 subtract 3 from both sides 1/6x = 1x - 10 subtract 1x from both sides -5/6x = -10 divide both sides by -5/6 x = 12
x = 12
In the list of numbers, find the one that is a solution of the given equation. -4, 14, 1 x - 9 = 5
x = 14
1/5 (3x + 5) - 1/3 (x + 7) = 4 3/5x + 1 - 1/3x - 2 1/3 = 4 combine like terms 4/15x - 1 1/3 = 4 LCD = 15 (15)(4/15x) + (15)(-1 1/3) + (15)(4) 4x - 20 = 60 add 20 to both sides 4x = 80 divide both sides by 4 x = 20
x = 20
1/9 (x + 2) = 2x - 2 (4 - x) - 7 1/6x + 1/3 = 2x - 8 + 2x - 7 combine like terms 1/6x + 1/3 = 4x - 15 subtract 1/3 from both sides 1/6x = 4x - 15 1/3 subtract 4x from both sides -3 5/6x = -15 1/3 divide both sides by -3 5/6 x = 4
x = 4
Which inequality is the solution for -3x < 6? a. x > -2 b. x ≥ -2 c. x ≤ -2 d. x < -2
x > -2
What is the word phrase that translates to x ≥ 3? a. x is greater than or equal to 3 b. x is less than or equal to 3 c. x is less than 3 d. x is greater than 3
x is greater than or equal to 3
Solving an Equation Using the Multiplication Property of Equality
1.) Multiply or divide both sides of the equation by the same number to get the variable x on a side of the equation by itself. - if x is being multiplied by a number, use division - if x is being divided by a number, use multiplication 2.) Simplify, if needed, by combining like terms 3.) Check your solution
Translating Words to Equations: When solving word problems, it is important to break down the problem to understand it.
1.) Read the word problem carefully to get an overview. 2.) Determine what information you will need to solve the problem. 3.) Draw a sketch or make a table. Label it with the known information.
To Determine if a Given Value is a Solution:
1.) Substitute the given value into the equation. 2.) Simplify each side of the equation according to the order of operations. 3.) If the result is a true statement, then that value is a solution.
To solve an equation, reverse operations are often needed.
1.) The reverse operation of addition is subtraction. 2.) The reverse operation of subtraction is addition.
2x - 5x = -12 combine like terms -3x = -12 divide both sides by -3 x = 4
2(4) - 5(4) = -12 8 - 20 = -12 -12 = -12
20 = -4x divide both sides by -4 -5 = x
20 = -4(-5) 20 = 20
7x = 14 divide both sides by 7 x = 2
7(2) = 14 14 = 14
*NOTE
An equation may have one solution, more than one solution, or no solution.
Which of the following statements about the Multiplication Property of Equality is FALSE? a. The Multiplication Property of Equality also applies to division. b. In symbols, for real numbers a, b, and c with c ≠ 0, if a = b, then ca = cb c. In symbols, for real numbers a, b, and c with c ≠ 0, if a = b, then a/c = b/c d. Both side of the equation can be multiplied by the same number c, and c can be any real number.
Both side of the equation can be multiplied by the same number c, and c can be any real number.
Is -1 a solution for the equation 2x + 6 = -1 ? 2(-1) + 6 = -1 -2 + 6 = 4
No
Is 27 a solution of x - 11 = 14 ?
No 27 - 11 = 16 16 ≠ 14
Is 4.3 a solution of 3x2 - 2 = 51.59 ?
No 3(4.3)2 - 2 = 51.59 3(18.49) - 2 55.47 - 2 53.47 ≠ 51.59
Is 4 a solution of x - 8 = 2 ?
No 4 - 8 = -4 -4 ≠ 2
6x2 - 3 = 4 (2 is squared) Ax + B = C
No, because x is squared *The variable in a linear equation cannot have an exponent greater than 1.
What would be the first calculation performed in order to solve the equation 2 - 8 = 2x + 9 ? a. Simplify the left side of the equation by combining 2 - 8 b. Add (-8) to both sides c. Simplify the right side of the equation by combining 2x + 9 d. Subtract 9 from both sides
Simplify the left side of the equation by combining 2 - 8
*NOTE
Since division can be performed by multiplying by the reciprocal, this property works for division as well. a, b, and c with c ≠ 0, if a = b, then a/c = b/c
Which is NOT a step to solving an equation using the Addition Property of Equality? a. Substitute numbers into the equation to find one that works. b. Once the value of x has been found, substitute the resulting value of x back into the original equation and simplify each side. c. Simplify each side, if needed, by combining like terms. d. Add or subtract the same number from both sides of the equation to get the variable x on a side of the equation by itself.
Substitute numbers into the equation to find one that works.
Choose the word problem that can be represented by an equation in one variable. a. The annual rainfall in Springfield is 3 inches more than the rainfall in Summerville. b. The larger of two numbers is two more than four times the smaller number. c. The winning soccer team earned 2 more than the other team's goals. The winning team earned 6 goals. d. The age of one child is six more than twice the youngest child's age.
The winning soccer team earned 2 more than the other team's goals. The winning team earned 6 goals.
Is 8 a solution of 2x + 9(x - 4) = 52 ?
Yes 2(8) + 9(8) + 9(-4) = 52 16 + 72 - 36 = 52 88 - 36 = 52 52 = 52
Is 13/5 a solution of 2x + 3x + 3 = 16 ?
Yes 5 1/5 + 7 4/5 + 3 = 16 13 + 3 = 16 16 = 16
Is 4 a solution of 6(x - 9) = -30 ?
Yes 6(4) + 6(-9) = -30 24 - 54 = -30 -30 = -30
Which of the following are reverse operations? a. subtraction and division b. addition and subtraction c. addition and multiplication d. multiplication and division
addition and subtraction, and multiplication and division
The sum of a number and twenty is negative eleven.
n + 20 = -11
Which of the following equations is equivalent to 2 + x = 10 ? a. x = 8 b. 2x = 10 c. 2 - x = 10 d. 2 = -x - 10
x = 8