Multi-Step Inequalities 2-4 Solving Two-Step and Multi-Step Inequalities Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1

Warm Up Solve each equation. 1. 2x – 5 = –17 2. –6 14 Solve each inequality and graph the solutions. 3. 5 < t + 9 t > –4 4. a ≤ –8

Objective Solve inequalities that contain more than one operation.

Inequalities that contain more than one operation require more than one step to solve. Use inverse operations to undo the operations in the inequality one at a time.

Example 1A: Solving Multi-Step Inequalities Solve the inequality and graph the solutions. 45 + 2b > 61 Since 45 is added to 2b, subtract 45 from both sides to undo the addition. 45 + 2b > 61 –45 –45 2b > 16 Since b is multiplied by 2, divide both sides by 2 to undo the multiplication. b > 8 2 4 6 8 10 12 14 16 18 20

Example 1B: Solving Multi-Step Inequalities Solve the inequality and graph the solutions. 8 – 3y ≥ 29 Since 8 is added to –3y, subtract 8 from both sides to undo the addition. 8 – 3y ≥ 29 –8 –8 –3y ≥ 21 Since y is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤. y ≤ –7 –10 –8 –6 –4 –2 2 4 6 8 10 –7

Solve the inequality and graph the solutions. Check It Out! Example 1a Solve the inequality and graph the solutions. –12 ≥ 3x + 6 Since 6 is added to 3x, subtract 6 from both sides to undo the addition. –12 ≥ 3x + 6 – 6 – 6 –18 ≥ 3x Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. –6 ≥ x –10 –8 –6 –4 –2 2 4 6 8 10

Solve the inequality and graph the solutions. Check It Out! Example 1b Solve the inequality and graph the solutions. Since x is divided by –2, multiply both sides by –2 to undo the division. Change > to <. –5 –5 x + 5 < –6 Since 5 is added to x, subtract 5 from both sides to undo the addition. x < –11 –20 –12 –8 –4 –16 –11

Solve the inequality and graph the solutions. Check It Out! Example 1c Solve the inequality and graph the solutions. Since 1 – 2n is divided by 3, multiply both sides by 3 to undo the division. 1 – 2n ≥ 21 Since 1 is added to –2n, subtract 1 from both sides to undo the addition. –1 –1 –2n ≥ 20 Since n is multiplied by –2, divide both sides by –2 to undo the multiplication. Change ≥ to ≤. n ≤ –10 –10 –20 –12 –8 –4 –16

To solve more complicated inequalities, you may first need to simplify the expressions on one or both sides by using the order of operations, combining like terms, or using the Distributive Property.

Example 2A: Simplifying Before Solving Inequalities Solve the inequality and graph the solutions. 2 – (–10) > –4t 12 > –4t Combine like terms. Since t is multiplied by –4, divide both sides by –4 to undo the multiplication. Change > to <. –3 < t (or t > –3) –3 –10 –8 –6 –4 –2 2 4 6 8 10

Example 2B: Simplifying Before Solving Inequalities Solve the inequality and graph the solutions. –4(2 – x) ≤ 8 –4(2 – x) ≤ 8 Distribute –4 on the left side. –4(2) – 4(–x) ≤ 8 Since –8 is added to 4x, add 8 to both sides. –8 + 4x ≤ 8 +8 +8 4x ≤ 16 Since x is multiplied by 4, divide both sides by 4 to undo the multiplication. x ≤ 4 –10 –8 –6 –4 –2 2 4 6 8 10

Solve the inequality and graph the solutions. Check It Out! Example 2a Solve the inequality and graph the solutions. 2m + 5 > 52 Simplify 52. 2m + 5 > 25 – 5 > – 5 Since 5 is added to 2m, subtract 5 from both sides to undo the addition. 2m > 20 m > 10 Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. 2 4 6 8 10 12 14 16 18 20

Solve the inequality and graph the solutions. Check It Out! Example 2b Solve the inequality and graph the solutions. 3 + 2(x + 4) > 3 Distribute 2 on the left side. 3 + 2(x + 4) > 3 3 + 2x + 8 > 3 Combine like terms. 2x + 11 > 3 Since 11 is added to 2x, subtract 11 from both sides to undo the addition. – 11 – 11 2x > –8 Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. x > –4 –10 –8 –6 –4 –2 2 4 6 8 10

Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. 13 – 2x ≥ 21 x ≤ –4 2. –11 + 2 < 3p p > –3 3. 23 < –2(3 – t) t > 7

Lesson Quiz: Part II 5. A video store has two movie rental plans. Plan A includes a $25 membership fee plus $1.25 for each movie rental. Plan B costs $40 for unlimited movie rentals. For what number of movie rentals is plan B less than plan A? more than 12 movies