Linear equations & Inequalities

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

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

Two more than two times a number is five.

2 + 2n = 5

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

3(7) = 21 21 = 21

3.5 times a number is seventeen.

3.5n = 17

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

3n = 8 + 5n

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

7(2) = 14 14 = 14

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

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

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

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.

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.

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

Solve for r. C = 2πr divide both sides by 2π C/2π = r

C/2π = r

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.

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)

5 - 4x ≥ -7 subtract 5 from both sides -4x ≥ -12 divide both sides by -4 change the direction of the inequality x ≤ 3

<-----● (3)

*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.

Formula

is an equation in which variables are used to describe a relationship.

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.

-15x = -195 divide both sides by -15 x = 13

-15(13) = -195 -195 = -195

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

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 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.

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

Solve for h. r = 1/5hy divide both sides by y 5r/y = hy/y 5r/y = h

5r/y = h

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

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"

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)

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.

Solving equations with parentheses

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

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 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.

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.

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

Linear Inequality

contains a single variable on either side of the inequality symbol. x > 3 3x - 5 < 8 4x - 6 ≥ 8x + 12

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.

Solution of an Inequality

is any number that makes the inequality true.

Variable

is a letter or symbol that represents an unknown quantity.

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

Thirteen minus four times a number is thirteen.

13 - 4n = 13

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

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

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.

*NOTE

If the decimals are tenths, multiply by 10; if the decimals are hundredths, multiply by 100, etc...

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.

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

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.

Solving an Inequality

When we solve an inequality, we are finding all the values that make the inequality true.

2x + 3 = 1 Ax + B = C

linear equation

The sum of a number and twenty is negative eleven.

n + 20 = -11

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

n + 50 = 188

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

n - 2 = 21

A number decreased by six is seventeen.

n - 6 = 17

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

14 = x - 7 add 7 to both sides 21 = x

14 = 21 - 7 14 = 14

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

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

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

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

-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

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

-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

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

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

2x + y = 92

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

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.

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.

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.

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.

Solving equations in the form Ax + B = C

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

Equivalent Equation

equations that have exactly the same solutions.

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)

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

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

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

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

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

9 + 2 = 11 11 = 11

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

*NOTE

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

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

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 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).

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.

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 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 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

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.

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

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

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

x = 14