Solving Multi-Step Inequalities Section 2.4

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

Solve the inequality and graph the solutions b > 61 –45 2b > 16 b > Since 45 is added to 2b, subtract 45 from both sides to undo the addition. Since b is multiplied by 2, divide both sides by 2 to undo the multiplication.

8 – 3y ≥ 29 –8 –3y ≥ 21 y ≤ –7 Since 8 is added to –3y, subtract 8 from both sides to undo the addition. Since y is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤. –10 –8 –6–4 – –7 Solve the inequality and graph the solutions.

–12 ≥ 3x + 6 – 6 –18 ≥ 3x –6 ≥ x Since 6 is added to 3x, subtract 6 from both sides to undo the addition. Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. –10 –8 –6–4 –

Solve the inequality and graph the solutions. x < –11 –5 x + 5 < –6 –20 –12–8–4 –16 0 –11 Since x is divided by –2, multiply both sides by –2 to undo the division. Change > to <. Since 5 is added to x, subtract 5 from both sides to undo the addition.

Solve the inequality and graph the solutions. 1 – 2n ≥ 21 –1 –2n ≥ 20 n ≤ –10 Since 1 – 2n is divided by 3, multiply both sides by 3 to undo the division. Since 1 is added to –2n, subtract 1 from both sides to undo the addition. Since n is multiplied by –2, divide both sides by –2 to undo the multiplication. Change ≥ to ≤. –10 –20 –12–8–4 –16 0

Solve the inequality and graph the solutions. 4f + 3 > 2 –3 4f > –1 Multiply both sides by 6, the LCD of the fractions. Distribute 6 on the left side. Since 3 is added to 4f, subtract 3 from both sides to undo the addition.

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

Solve the inequality and graph the solutions. 5 < 3x – < 3x Multiply both sides by 8, the LCD of the fractions. Distribute 8 on the right side. Since 2 is subtracted from 3x, add 2 to both sides to undo the subtraction.

Example 3 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 is 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 <  m

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

Example 4 The average of Jim ’ s two test scores must be at least 90 to make an A in the class. Jim got a 95 on his first test. What grades can Jim get on his second test to make an A in the class? Let x represent the test score needed. The average score is the sum of each score divided by 2. First test score plus second test score divided by number of scores is greater than or equal to total score (95 + x) x)  2 ≥ 90

The score on the second test must be 85 or higher. Since 95 is added to x, subtract 95 from both sides to undo the addition x ≥ 180 –95 x ≥ 85 Since 95 + x is divided by 2, multiply both sides by 2 to undo the division.