Linear equations & Inequalities

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

-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

-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

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/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

x/3 = -15 divide both sides by 3 x = -45

-45/3 = -15 -15 = -15

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

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

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

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 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

20 = -4x divide both sides by -4 -5 = x

20 = -4(-5) 20 = 20

Which of the following is an equation? a. 2x + y = 92 b. 5x - 4y c. 28x / 9 d. 2x

2x + y = 92

3x = 21 divide both sides by 3 x = 7

3(7) = 21 21 = 21

3.5 times a number is seventeen.

3.5n = 17

x + 2 = 11 subtract 2 from both sides x = 9

9 + 2 = 11 11 = 11

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 = 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

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

-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

-6 = x + 2 subtract 2 from both sides -8 = x

-6 = -8 + 2 -6 = -6

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.

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.

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.

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

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

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)

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

The sum of a number and fifty is one hundred eighty-eight.

n + 50 = 188

Triple a number is equal to eight more than five times the number.

3n = 8 + 5n

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 - 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

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

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 + 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

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

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

7x = 14 divide both sides by 7 x = 2

7(2) = 14 14 = 14

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

The difference between a number and two is twenty-one.

n - 2 = 21

*NOTE

Addition and subtraction "undo" each other, meaning that adding and subtracting the same number result in no change.

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.

*NOTE

An equation may have one solution, more than one solution, or no solution.

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

Solving equations with parentheses

For all real numbers a, b, and c, a(b + c) = ab + ac

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

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.

*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

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).

A number decreased by six is seventeen.

n - 6 = 17

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.

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.

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.

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

Is -1 a solution for the equation 2x + 6 = -1 ? 2(-1) + 6 = -1 -2 + 6 = 4

No

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

*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

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.

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.

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

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.

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.

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.

Determine whether the equation is a linear equation. -4x + 6 = 2

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

Solving equations in the form Ax + B = C

You must use both the Addition Property of Equality and the Multiplication Property of Equality together.

Solution

a solution of an equation is the number(s) that, when substituted for the variable(s), makes the equation true.

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

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 ( = )

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

2x + 3 = 1 Ax + B = C

linear equation

The sum of a number and twenty is negative eleven.

n + 20 = -11

In the list of numbers, find the one that is a solution of the given equation. -4, 14, 1 x - 9 = 5

x = 14

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