Name ______Date ______Period ______

Unit 3 – Inequalities

Study Guide

Introduction to Inequalities

  1. Match each phrase on the left with it’s corresponding symbol on the right.

Phrase
“…is fewer than…”
“…is no more than…”
“…is at least…”
“…is at most…”
“…is more than…"
“…is no less than…”
Symbol
≤
≥
  1. Tell whether -8 is a solution of each inequality.

a. x + 15 ≥ -10b.

  1. Write an inequality for each graph below. Then, write a second, equivalent inequality.

a. b.

  1. Write the word sentence as an inequality.
  2. A number y is no more than -8
  1. A number w added to 2.3 is more than 18.
  1. A number t multiplied by -4 is at least .
  1. A number b minus 4.2 is less than -7.5.

One-Step Inequalities

Directions: Solve and graph each inequality.

  1. 3 + x < 10
  1. 7x ≥ 56
  1. 24 ≤ 9 – x
  1. John is selling cupcakes to raise money for charity. For each cupcake sold, John earns $.80. How many cupcakes would John have to sell if he wants to donate at least $50.00 to charity? Write an inequality and solve. Be sure to express your answer with the correct units.

Let x =

  1. A small airplane can hold 44 passengers. Fifteen passengers board the plane. Write and solve an inequality that represents the additional number of passengers that can board the plane.

Let x =

b. Can 30 more passengers board the plane? Explain.

Two-Step Inequalities

Directions: Solve and graph each inequality.

  1. 4x – 2 ≤ 12
  1. 9 – 3x ≥ -21
  1. The first jump in a unicycle high-jump contest is 10 centimeters. The bar is raised 2 centimeters after each jump. Write and solve an inequality to find the number of additional jumps needed to meet or exceed the goal of clearing a height of 26 centimeters. Be sure to express your answer with the correct units.

Let x =

  1. A scuba diver is at an elevation of -38 feet. The diver starts moving at a rate of -12 feet per minute. Write and solve an inequality that represents how long it will take the diver to reach an elevation deeper than -200 feet. Be sure to express your answer with the correct units.

Let x =
Inequalities with Limited Solution Sets

If possible, find the set of solutions that would make both inequalities true. Graph the solution and write it as a compound inequality.

  1. 4x - 3 < 21 and 9x ≥ 27
  1. -3x + 2 > 4 and 12 – x < 10
  1. 2x – 5 ≥ 11 and
  1. and 2 – 4x ≥ -18

  1. A video game gives you 100 seconds to complete the level and move to the next. You are halfway through the level after 55 seconds.
  1. Write and solve an inequality to find out how much time you have left to complete the level.
  1. You will receive a time bonus if you finish in 70 seconds or less. Write and solve an inequality to find how much time you have left to earn a time bonus.
  1. You finish the game in another 32 seconds. Do you earn a time bonus? Do you move to the next level? Explain
  1. An isosceles triangle has a base of 5 centimeters and legs x centimeters long. The perimeter is no more than 30 centimeters. Write and solve an inequality to find the possible values of x.
  1. The perimeter of the rectangle is at least 12 inches. The area is no more than 27 inches. Write and solve an inequality for each condition. Give two possible values for x.

Answer Key

< “…is fewer than…”
≤ “…is no more than…”
≥ “…is at least…”
≤ “…is at most…”
> “…is more than…"
≥ “…is no less than…”
  1. a. Yes, -8 is a solution

b. No, -8 is not a solution

  1. a. x > 12

b. x ≤ -4

  1. a. y ≤ -8
  1. 2.3 + w > 18
  2. -4t ≥ -2/5
  3. b – 4.2 < -7.5
  1. x < 7
  2. x ≥ 8
  3. 15 ≥ x
  4. x > -36
  5. .80x ≥ 50

John has to sell at least 63 cupcakes

  1. x + 15 ≤ 44

No more than 29 passengers can board the plane.

b. No, only 29 or fewer passengers can board the plane and 30 > 29.

  1. x ≤ 3.5
  2. x ≤ 10
  3. x > 92
  4. x < 57
  5. 10 + 2x ≥ 26 ; At least 8 jumps are needed to meet or exceed the goal
  6. -12x + (-38) ≤ -200 ; The diver will have to dive for at least 13.5 seconds to reach a depth of -200 feet.
  7. 3 ≤ x < 6
  8. No solutions would make both of the inequalities true
  9. x > 24
  10. -22 < x ≤ 5
  11. a. x + 55 ≤ 100
  1. x + 55 ≤ 70
  2. No, you do not earn a time bonus because 87 seconds is not less than the required 70 seconds.

Yes, you do move on to the next level because 87 seconds is less than the required 100 seconds.

  1. 2x + 5 ≤ 30 ; x must be no more than 12.5 centimeters
  2. 2x + 6 ≥ 12 and 3x ≤ 27 ; Any values greater than or equal to 3 but less than or equal to 9 would make this true. Sample answers: 3, 4, 5.7, 8.1, 9