Solve inequalities that contain variable terms on both sides.

1. Objective Solve inequalities that contain variable terms on both sides.

2. y ≤ 4y + 18 –y –y 0 ≤ 3y + 18 –18 – 18 –18 ≤ 3y –8 –10 –6 –4 0 2 4 6 8 10 –2 Example 1A: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. y ≤ 4y + 18 To collect the variable terms on one side, subtract y from both sides. Since 18 is added to 3y, subtract 18 from both sides to undo the addition. Since y is multiplied by 3, divide both sides by 3 to undo the multiplication. –6 ≤ y (or y –6)

3. –2m –2m 2m – 3 < + 6 + 3 + 3 2m < 9 4 5 6 Example 1B: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. 4m – 3 < 2m + 6 To collect the variable terms on one side, subtract 2m from both sides. Since 3 is subtracted from 2m, add 3 to both sides to undo the subtraction Since m is multiplied by 2, divide both sides by 2 to undo the multiplication.

4. 4x ≥ 7x + 6 –7x –7x x ≤ –2 –8 –10 –6 –4 0 2 4 6 8 10 –2 Check It Out! Example 1a Solve the inequality and graph the solutions. 4x ≥ 7x + 6 To collect the variable terms on one side, subtract 7x from both sides. –3x ≥ 6 Since x is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤.

5. 5t + 1 < –2t – 6 +2t +2t 7t + 1 < –6 – 1 < –1 7t < –7 7t < –7 7 7 t < –1 –4 –1 5 –3 –2 0 1 2 3 4 –5 Check It Out! Example 1b Solve the inequality and graph the solutions. 5t + 1 < –2t – 6 To collect the variable terms on one side, add 2t to both sides. Since 1 is added to 7t, subtract 1 from both sides to undo the addition. Since t is multiplied by 7, divide both sides by 7 to undo the multiplication.

6. –2k –2k –3 –3 Example 3A: Simplify Each Side Before Solving Solve the inequality and graph the solutions. 2(k – 3) > 6 + 3k – 3 Distribute 2 on the left side of the inequality. 2(k – 3) > 3 + 3k 2k+ 2(–3)> 3 + 3k 2k –6 > 3 + 3k To collect the variable terms, subtract 2k from both sides. –6 > 3 + k Since 3 is added to k, subtract 3 from both sides to undo the addition. –9 > k

7. –12 –9 –6 –3 0 3 Example 3A Continued –9 > k

8. +6 +6 + 5r +5r Check It Out! Example 3a Solve the inequality and graph the solutions. 5(2 – r) ≥ 3(r – 2) Distribute 5 on the left side of the inequality and distribute 3 on the right side of the inequality. 5(2 – r) ≥ 3(r – 2) 5(2) – 5(r) ≥ 3(r) + 3(–2) Since 6 is subtracted from 3r, add 6 to both sides to undo the subtraction. 10 – 5r ≥ 3r – 6 16 − 5r ≥ 3r Since 5r is subtracted from 16 add 5r to both sides to undo the subtraction. 16 ≥ 8r

9. –6 –4 –2 0 2 4 Check It Out! Example 3a Continued 16 ≥ 8r Since r is multiplied by 8, divide both sides by 8 to undo the multiplication. 2 ≥ r

10. + 0.3 + 0.3 –0.3x –0.3x Check It Out! Example 3b Solve the inequality and graph the solutions. 0.5x – 0.3 + 1.9x < 0.3x + 6 2.4x –0.3 < 0.3x + 6 Simplify. Since 0.3 is subtracted from 2.4x, add 0.3 to both sides. 2.4x –0.3 < 0.3x + 6 2.4x < 0.3x + 6.3 Since 0.3x is added to 6.3, subtract 0.3x from both sides. 2.1x < 6.3 Since x is multiplied by 2.1, divide both sides by 2.1. x < 3

11. –4 –1 5 –3 –2 0 1 2 3 4 –5 Check It Out! Example 3b Continued x < 3

12. Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. t < 5t + 24 t > –6

13. Lesson Quiz: Part I Solve each inequality and graph the solutions. 3. 4b + 4(1 – b) > b – 9 b < 13