Objective Solve inequalities that contain more than one operation.

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

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

daily cost at We Got Wheels Example 3: Application To rent a certain vehicle, Rent-A-Ride charges $55.00 per day with unlimited miles. The cost of renting a similar vehicle at We Got Wheels is $38.00 per day plus $0.20 per mile. For what number of miles in the cost at Rent-A-Ride less than the cost at We Got Wheels? Let m represent the number of miles. The cost for Rent-A-Ride should be less than that of We Got Wheels. Cost at Rent-A-Ride must be less than daily cost at We Got Wheels plus $0.20 per mile times # of miles. 55 < 38 + 0.20  m

Example 3 Continued 55 < 38 + 0.20m Since 38 is added to 0.20m, subtract 38 from both sides to undo the addition. –38 –38 55 < 38 + 0.20m 17 < 0.20m Since m is multiplied by 0.20, divide both sides by 0.20 to undo the multiplication. 85 < m Rent-A-Ride costs less when the number of miles is more than 85.

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 4.

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