1. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview

2. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Warm Up Solve. 1. 6x + 36 = 2x 2. 4x – 13 = 15 + 5x 3. 5(x – 3) = 2x + 3 4. + x = x = –9 x = –28 x = 6 7 8 13 16 1 x = –

3. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities 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

4. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities 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.

5. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Solve and graph. Additional Example 1A: Solving Two-Step Inequalities 4x + 1 > 13 – 1 – 1Since 1 is added to 4x, subtract 1 from both sides. 4x > 12 4x4x 4 > 12 4 Since x is multiplied by 4, divide both sides by 4. x > 3 1 2 3 4 5 6 7

6. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Additional 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 2 > 3 ? Substitute 2 for x. So 2 is not a solution. 

7. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed. Remember!

8. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Additional Example 1B: Solving Two-Step Inequalities –9x + 7  25 – 7 – 7Subtract 7 from both sides. –9x  18 –9x –9  18 –9 Divide each side by –9; change  to . x  –2x  –2 -6 -5 -4 -3 -2 -1 0 Solve and graph.

9. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Solve and graph. Check It Out! Example 1A 5x + 2 > 12 – 2 – 2Subtract 2 from both sides. 5x > 10 5x5x 5 > 10 5 Divide both sides by 5. x > 2 1 2 3 4 5 6 7

10. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Check It Out! Example 1A Continued Check x > 2 4 > 2 ? Substitute 4 for x. According to the graph 4 should be a solution and 1 should not be a solution. So 4 is a solution. x > 2 1 > 2 ? Substitute 1 for x. So 1 is not a solution. 

11. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities –4x + 2  18 – 2 – 2Subtract 2 from both sides. –4x  16 –4x –4  16 –4 Divide each side by –4; change  to . x  –4x  –4 -6 -5 -4 -3 -2 -1 0 Check It Out! Example 1B

12. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Additional 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. 2x 5 3 4 9 10 +  2x2x 5 3 4 9 10 20 ( + )  20 ( ) 2x2x 5 3 4 9 10 20 ( ) + 20 ( )  20 ( ) 2x2x 5 3 4 9 10 Distributive Property.

13. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Additional Example 2 Continued x  3 8  8x8x 8 3 8 Since x multiplied by 8, divide both sides by 8. 8x  3 0 1 3 8

14. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Check 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 +  3x 5 1 4 5 10 +  3x3x 5 1 4 5 10 20 ( + )  20 ( ) 3x3x 5 1 4 5 10 20 ( ) + 20 ( )  20 ( ) 3x3x 5 1 4 5 10 Distributive Property.

15. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Check It Out! Example 2 Continued x  5 12  12x 12 5 Since x is multiplied by 12, divide both sides by 12. 12x  5 0 5 12

16. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Additional 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

17. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Additional Example 3 Continued – 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 1.10x 1.10 55 1.10 >

18. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Check 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

19. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities – 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 2.25x 2.25 45 2.25 > Check It Out! Example 3 Continued

20. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Lesson 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 > –16 2 3 1 4 1 2 w w  3 8 1 2 3 4 5 6 7-10 -9 -8 -7 -6 -5 -4-18 -17 -16 -15 -14 -13 -12 0 3 8

21. Evaluating Algebraic Expressions 3-8Solving Two-Step Inequalities Lesson 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