Solving TwoStep and 3 4 MultiStep Inequalities Warm

Solving Two-Step and 3 -4 Multi-Step Inequalities Warm Up Solve each equation. 1. 2 x – 5 = – 17 – 6 2. 14 Solve each inequality and graph the solutions. 3. 5 < t + 9 t > – 4 4. Holt Algebra 1 a ≤ – 8

Solving Two-Step and 3 -4 Multi-Step Inequalities Objective Solve inequalities that contain more than one operation. Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step 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. Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step Inequalities Directions: Solve the inequality and graph the solutions. Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 1 45 + 2 b > 61 – 45 2 b > 16 b>8 0 2 4 Holt Algebra 1 6 Since 45 is added to 2 b, 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 10 12 14 16 18 20

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 2 8 – 3 y ≥ 29 – 8 Since 8 is added to – 3 y, subtract 8 from both sides to undo the addition. – 3 y ≥ 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 Holt Algebra 1 0 2 4 6 8 10

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 3 – 12 ≥ 3 x + 6 – 6 Since 6 is added to 3 x, subtract 6 from both sides to undo the addition. – 18 ≥ 3 x Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. – 6 ≥ x – 10 – 8 – 6 – 4 – 2 Holt Algebra 1 0 2 4 6 8 10

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 4 Since x is divided by – 2, multiply both sides by – 2 to undo the division. Change > to <. x + 5 < – 6 – 5 Since 5 is added to x, subtract 5 from both sides to undo the addition. x < – 11 – 20 – 16 Holt Algebra 1 – 12 – 8 – 4 0

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 5 1 – 2 n ≥ 21 – 1 – 2 n ≥ 20 Since 1 – 2 n is divided by 3, multiply both sides by 3 to undo the division. Since 1 is added to − 2 n, 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 ≤. n ≤ – 10 – 20 Holt Algebra 1 – 16 – 12 – 8 – 4 0

Solving Two-Step and 3 -4 Multi-Step Inequalities 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. Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 6 2 – (– 10) > – 4 t 12 > – 4 t 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 Holt Algebra 1 0 2 4 6 8 10

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 7 – 4(2 – x) ≤ 8 − 4(2) − 4(−x) ≤ 8 – 8 + 4 x ≤ 8 +8 +8 4 x ≤ 16 Distribute – 4 on the left side. Since – 8 is added to 4 x, add 8 to both sides. Since x is multiplied by 4, divide both sides by 4 to undo the multiplication. x≤ 4 – 10 – 8 – 6 – 4 – 2 Holt Algebra 1 0 2 4 6 8 10

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 8 Multiply both sides by 6, the LCD of the fractions. Distribute 6 on the left side. 4 f + 3 > 2 – 3 4 f Holt Algebra 1 > – 1 Since 3 is added to 4 f, subtract 3 from both sides to undo the addition.

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 8 Continued 4 f > – 1 Since f is multiplied by 4, divide both sides by 4 to undo the multiplication. 0 Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 9 2 m + 5 > 52 2 m + 5 > 25 – 5>– 5 2 m > 20 m > 10 0 2 4 6 Holt Algebra 1 Simplify 52. Since 5 is added to 2 m, subtract 5 from both sides to undo the addition. Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. 8 10 12 14 16 18 20

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 10 3 + 2(x + 4) > 3 Distribute 2 on the left side. 3 + 2(x + 4) > 3 3 + 2 x + 8 > 3 Combine like terms. Since 11 is added to 2 x, subtract 11 from both sides to undo the addition. 2 x + 11 > 3 – 11 2 x > – 8 Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. x > – 4 – 10 – 8 – 6 – 4 – 2 Holt Algebra 1 0 2 4 6 8 10

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 11 Multiply both sides by 8, the LCD of the fractions. Distribute 8 on the right side. 5 < 3 x – 2 +2 +2 7 < 3 x Holt Algebra 1 Since 2 is subtracted from 3 x, add 2 to both sides to undo the subtraction.

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 11 Continued Solve the inequality and graph the solutions. 7 < 3 x Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. 0 2 Holt Algebra 1 4 6 8 10

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 12: 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-ARide must be less than 55 < Holt Algebra 1 daily cost at We Got Wheels 38 plus + $0. 20 per mile 0. 20 times # of miles. m

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 12 Continued 55 < 38 + 0. 20 m Since 38 is added to 0. 20 m, subtract 8 55 < 38 + 0. 20 m from both sides to undo the addition. – 38 17 < 0. 20 m 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. Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 12 Continued Check the endpoint, 85. 55 = 38 + 0. 20 m Check a number greater than 85. 55 < 38 + 0. 20 m 55 38 + 0. 20(85) 55 < 38 + 0. 20(90) 55 55 38 + 17 55 < 38 + 18 55 < 56 Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 13 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 (95 Holt Algebra 1 + second test score x) divided by number of scores 2 is greater than or equal to ≥ total score 90

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 13 Continued Since 95 + x is divided by 2, multiply both sides by 2 to undo the division. 95 + x ≥ 180 – 95 Since 95 is added to x, subtract 95 from both sides to undo the addition. x ≥ 85 The score on the second test must be 85 or higher. Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step Inequalities Example 13 Continued Check the end point, 85. Check a number greater than 85. 90 90 90 Holt Algebra 1 90 90. 5 ≥ 90

Solving Two-Step and 3 -4 Multi-Step Inequalities Lesson Summary: Part I Solve each inequality and graph the solutions. 1. 13 – 2 x ≥ 21 x ≤ – 4 2. – 11 + 2 < 3 p p > – 3 3. 23 < – 2(3 – t) t>7 4. Holt Algebra 1

Solving Two-Step and 3 -4 Multi-Step Inequalities Lesson Summary: 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 Holt Algebra 1