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

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

Example 2C Continued 4f > –1 Since f is multiplied by 4, divide both sides by 4 to undo the multiplication.

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

Check It Out! Example 2c Solve the inequality and graph the solutions. Multiply both sides by 8, the LCD of the fractions. Distribute 8 on the right side. 5 < 3x – 2 Since 2 is subtracted from 3x, add 2 to both sides to undo the subtraction. +2 + 2 7 < 3x

Check It Out! Example 2c Continued Solve the inequality and graph the solutions. 7 < 3x Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. 4 6 8 2 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 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 < 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.

  Example 3 Continued Check Check the endpoint, 85. 55 38 + 17 55 55 55 38 + 17 55 55  55 = 38 + 0.20m 55 38 + 0.20(85) Check a number greater than 85. 55 < 38 + 18 55 < 56  55 < 38 + 0.20(90) 55 < 38 + 0.20m

is greater than or equal to Check It Out! Example 3 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)  2 ≥ 90

Check It Out! Example 3 Continued Since 95 + x is divided by 2, multiply both sides by 2 to undo the division. 95 + x ≥ 180 Since 95 is added to x, subtract 95 from both sides to undo the addition. –95 –95 x ≥ 85 The score on the second test must be 85 or higher.

Check It Out! Example 3 Continued Check the end point, 85. Check a number greater than 85. 90 90 90  90.5 ≥ 90 

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