Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Warm Up Warm Up California...

Evaluating Algebraic Expressions

3-8 Solving Two-Step Inequalities

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California StandardsCalifornia Standards

Lesson PresentationLesson Presentation

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Warm UpSolve.

1. 6x + 36 = 2x

2. 4x – 13 = 15 + 5x

3. 5(x – 3) = 2x + 3

4. + x =

x = –9

x = –28

x = 678

1316

1 16x = –



AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.

California Standards



When you solved two-step equations, you used the order of operations in reverse to isolate the variable. You can use the same process when solving two-step inequalities.



Solve and graph.

Additional Example 1A: Solving Two-Step Inequalities

4x + 1 > 13

4x + 1 > 13 – 1 – 1 Since 1 is added to 4x,

subtract 1 from both sides. 4x > 124x4

> 124

Since x is multiplied by 4, divide both sides by 4.

x > 3

1 2 3 4 5 6 7


3-8 Solving Two-Step InequalitiesAdditional Example 1A Continued

Check

x > 3

4 > 3?Substitute 4 for x.

According to the graph 4 should be a solution and 2 should not be a solution.

So 4 is a solution.

x > 3


So 2 is not a solution.



If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed.

Remember!



Additional Example 1B: Solving Two-Step Inequalities

–9x + 7 25

–9x + 7 25

– 7 – 7 Subtract 7 from both sides.

–9x 18

–9x–9

18–9

Divide each side by –9; change to .

x –2-6 -5 -4 -3 -2 -1 0

Solve and graph.



Solve and graph.

Check It Out! Example 1A

5x + 2 > 12

5x + 2 > 12 – 2 – 2 Subtract 2 from both sides.

5x > 105x5

> 105

Divide both sides by 5.

x > 2 1 2 3 4 5 6 7


3-8 Solving Two-Step InequalitiesCheck It Out! Example 1A Continued

Check

x > 2


According to the graph 4 should be a solution and 1 should not be a solution.

So 4 is a solution.

x > 2


So 1 is not a solution.



–4x + 2 18

–4x + 2 18

– 2 – 2 Subtract 2 from both sides.

–4x 16

–4x–4

16–4

Divide each side by –4; change to .

x –4-6 -5 -4 -3 -2 -1 0

Check It Out! Example 1B


3-8 Solving Two-Step InequalitiesAdditional Example 2: Solving Inequalities That Contain

Fractions

Multiply by LCD, 20.

8x + 15 18

– 15 – 15 Since 15 is added to 8x,

subtract 15 from both sides.8x 3

Solve + and graph the solution.2x5

34

910

+ 2x5

34

910

20( + ) 20( )2x5

34

910

20( ) + 20( ) 20( )2x5

34

910 Distributive Property.


3-8 Solving Two-Step InequalitiesAdditional Example 2 Continued

x 38

8x8

38 Since x multiplied by 8, divide both

sides by 8.

8x 3

0 1

38


3-8 Solving Two-Step InequalitiesCheck It Out! Example 2

Multiply by LCD, 20.

12x + 5 10

– 5 – 5 Since 5 is added to 12x,

subtract 5 from both sides.

12x 5

Solve + 3x5

14

510

+ 3x5

14

510

20( + ) 20( )3x5

14

510

20( ) + 20( ) 20 ( )3x5

14

510 Distributive Property.


3-8 Solving Two-Step InequalitiesCheck It Out! Example 2 Continued

x 512

12x12

512 Since x is multiplied by 12,

divide both sides by 12.

12x 5

0 5 12


3-8 Solving Two-Step InequalitiesAdditional Example 3: School Application

A school’s Spanish club is selling bumper stickers. They bought 100 bumper stickers for $55, and they have to give the company 15 cents for every sticker sold. If they plan to sell each bumper sticker for $1.25, how many do they have to sell to make a profit?

In order for the Spanish club to make a profit, the revenue must be greater than the cost.

1.25x > 55 + 0.15x


3-8 Solving Two-Step InequalitiesAdditional Example 3 Continued

– 0.15x – 0.15x Subtract 0.15x from both sides.

1.10x > 55

x > 50

The Spanish club must sell more than 50 bumper stickers to make a profit.

Divide both sides by 1.10.

1.25x > 55 + 0.15x

1.10x1.10

551.10>


3-8 Solving Two-Step InequalitiesCheck It Out! Example 3

A school’s French club is selling bumper stickers. They bought 200 bumper stickers for $45, and they have to give the company 25 cents for every sticker sold. If they plan to sell each bumper sticker for $2.50, how many do they have to sell to make a profit?

In order for the French club to make a profit, the revenue must be greater than the cost.

2.5x > 45 + 0.25x



– 0.25x – 0.25x Subtract 0.25x from both sides.

2.25x > 45

x > 20

The French club must sell more than 20 bumper stickers to make a profit.

Divide both sides by 2.25.

2.5x > 45 + 0.25x

2.25x2.25

452.25>

Check It Out! Example 3 Continued


3-8 Solving Two-Step InequalitiesLesson Quiz: Part I

Solve and graph.

1. 4x – 6 > 10

2. 7x + 9 < 3x – 15

3. w – 3w < 32

4. w +

x < –6

x > 4

w > –1623

14

12

w 38

1 2 3 4 5 6 7

-10 -9 -8 -7 -6 -5 -4

-18 -17 -16 -15 -14 -13 -12

0 38


3-8 Solving Two-Step InequalitiesLesson Quiz: Part II

5. Antonio has budgeted an average of $45 a month for entertainment. For the first five months of the year he has spent $48, $39, $60, $48, and $33. How much can Antonio spend in the sixth month without exceeding his average budget?

no more than $42

Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Warm Up Warm Up California...

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