This document provides examples for solving two-step linear equations and inequalities. It begins with examples of solving two-step inequalities by using the reverse order of operations to isolate the variable. It then discusses multiplying or dividing both sides of an inequality by a negative number. Additional examples include solving inequalities containing fractions and a word problem about a school club selling bumper stickers.

1. Warm Up California Standards Lesson Presentation Preview

2. Warm Up Solve. 1. 6 x + 36 = 2 x 2. 4 x – 13 = 15 + 5 x 3. 5( x – 3) = 2 x + 3 4. + x = x = –9 x = –28 x = 6 7 8 13 16 1 16 x = –

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

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

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

9. Solve and graph. Check It Out! Example 1A 5 x + 2 > 12 5 x + 2 > 12 – 2 – 2 Subtract 2 from both sides. 5 x > 10 Divide both sides by 5. x > 2 5 x 5 > 10 5 1 2 3 4 5 6 7

10. 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. – 4 x + 2  18 – 4 x + 2  18 – 2 – 2 Subtract 2 from both sides. – 4 x  16 Divide each side by –4; change  to  . x  – 4 Check It Out! Example 1B – 4 x – 4  16 – 4 -6 -5 -4 -3 -2 -1 0

12. Additional Example 2: Solving Inequalities That Contain Fractions Multiply by LCD, 20. 8 x + 15  18 – 15 – 15 Since 15 is added to 8x, subtract 15 from both sides. 8 x  3 Solve +  and graph the solution. Distributive Property. 2x 5 3 4 9 10 +  2 x 5 3 4 9 10 20 ( + )  20 ( ) 2 x 5 3 4 9 10 20 ( ) + 20 ( )  20 ( ) 2 x 5 3 4 9 10

13. Additional Example 2 Continued 8 x  3 x  3 8  8 x 8 3 8 Since x multiplied by 8, divide both sides by 8. 0 1 3 8

14. Check It Out! Example 2 Multiply by LCD, 20. 12 x + 5  10 – 5 – 5 Since 5 is added to 12x, subtract 5 from both sides. 12 x  5 Distributive Property. Solve +  3x 5 1 4 5 10 +  3 x 5 1 4 5 10 20 ( + )  20 ( ) 3 x 5 1 4 5 10 20 ( ) + 20 ( )  20 ( ) 3 x 5 1 4 5 10

15. Check It Out! Example 2 Continued 12 x  5 x  5 12  12 x 12 5 12 Since x is multiplied by 12, divide both sides by 12. 0 5 12

16. 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. 1.25 x > 55 + 0.15 x

17. Additional Example 3 Continued – 0.15x – 0.15 x Subtract 0.15x from both sides. 1.10 x > 55 x > 50 The Spanish club must sell more than 50 bumper stickers to make a profit. Divide both sides by 1.10. 1.25 x > 55 + 0.15 x 1.10 x 1.10 55 1.10 >

18. 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. 2.5 x > 45 + 0.25 x

19. – 0.25x – 0.25 x Subtract 0.25x from both sides. 2.25 x > 45 x > 20 The French club must sell more than 20 bumper stickers to make a profit. Divide both sides by 2.25. 2.5 x > 45 + 0.25 x Check It Out! Example 3 Continued 2.25 x 2.25 45 2.25 >

20. Lesson Quiz: Part I Solve and graph. 1. 4 x – 6 > 10 2. 7 x + 9 < 3 x – 15 3. w – 3 w < 32 4. w +  x < –6 x > 4 w > –16 2 3 1 4 1 2 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. 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