Absolute Value Inequalities: Distance on a Number Line

Materials Needed

• Pencil and paper or a whiteboard
• Number line, ruler, or two pieces of string
• Optional: graphing tool or digital number-line app
• Timer

Learning Objectives

By the end of this 15-minute lesson, the learner will be able to:

• Explain that absolute value represents distance from zero.
• Solve one-variable absolute value inequalities such as |x - a| < b and |x - a| > b.
• Represent solutions using an inequality, interval notation, and a graph on a number line.
• Apply an absolute value inequality to a real-world situation.

Success Criteria

A successful solution will:

• Identify the center value and distance.
• Use the correct compound inequality or “or” statement.
• Use open circles for < or >, and closed circles for ≤ or ≥.
• Shade all and only the numbers that make the inequality true.

Introduction and Hook — 2 Minutes

Hook: Imagine that a package must weigh within 2 pounds of a target weight of 10 pounds. What weights are acceptable?

Explain that “within 2 pounds of 10” means the distance between the weight and 10 is less than or equal to 2:

|w - 10| ≤ 2

Ask: “Would 8 pounds be acceptable? What about 12 pounds? What about 7 pounds?”

Tell the learner: “Today we will use absolute value inequalities to describe distances and solve them algebraically and graphically.”

Body: I Do — Teacher Modeling — 4 Minutes

Key Idea

|x - a| means the distance between x and a on a number line.

Example 1: Less Than Means Between

Solve and graph:

|x - 3| < 4

1. The center is 3.
2. The distance is less than 4.
3. Therefore, x must be between 4 units below and 4 units above 3:

-4 < x - 3 < 4

Add 3 to every part:

-1 < x < 7

Graph: Put open circles at -1 and 7, then shade between them.

Meaning: The solution includes numbers less than 7 and greater than -1, but not the endpoints.

Example 2: Greater Than Means Outside

Solve and graph:

|x + 2| ≥ 5

1. Rewrite x + 2 as x - (-2). The center is -2.
2. The distance is at least 5, so x is 5 or more units away from -2.
3. Use two possibilities:

x + 2 ≤ -5 or x + 2 ≥ 5

Subtract 2 from both inequalities:

x ≤ -7 or x ≥ 3

Graph: Put closed circles at -7 and 3. Shade left of -7 and right of 3.

Memory tip:

• Less than: solutions stay inside the interval.
• Greater than: solutions go outside the interval.

Body: We Do — Guided Practice — 4 Minutes

Work through each problem together. Have the learner explain what the absolute value represents before solving.

Problem A

Solve and graph:

|x - 5| ≤ 2

Guide the learner with these questions:

1. What is the center? 5
2. What is the distance? 2
3. Should the answer be inside or outside? Inside
4. Are the endpoints included? Yes, because the symbol is ≤

Expected solution:

3 ≤ x ≤ 7

Graph with closed circles at 3 and 7, shading between them.

Problem B

Solve and graph:

|x + 1| > 3

Ask:

• What is the center? -1
• Should the solution be inside or outside? Outside
• Are the endpoints included? No, because the symbol is >

Expected solution:

x < -4 or x > 2

Graph with open circles at -4 and 2, shading outward.

Quick Check: Ask the learner to point to a number that belongs to each solution and a number that does not. For Problem B, 5 belongs, but 0 does not.

Body: You Do — Independent Application — 3 Minutes

Have the learner choose one of the following real-world problems, solve it, and graph the solution.

Choice 1: Temperature

A refrigerator should stay within 3°F of 38°F. Let t represent the temperature.

Write and solve an inequality for the acceptable temperatures.

Expected answer:

|t - 38| ≤ 3

35 ≤ t ≤ 41

The graph has closed circles at 35 and 41, with shading between them.

Choice 2: Distance from a Landmark

A cyclist wants to be more than 4 miles away from a starting point located at mile marker 6. Let d represent the cyclist’s position.

Write and solve an inequality.

Expected answer:

|d - 6| > 4

d < 2 or d > 10

The graph has open circles at 2 and 10, with shading outward.

Conclusion and Assessment — 2 Minutes

Exit Ticket

Solve and graph:

|x - 4| < 2

Expected answer:

2 < x < 6

Graph with open circles at 2 and 6, shading between them.

Recap

Ask the learner to complete these statements:

• Absolute value represents the distance from __________.
• An absolute value inequality with “less than” describes solutions __________ the endpoints.
• An absolute value inequality with “greater than” describes solutions __________ the endpoints.
• Open circles represent __________, while closed circles represent __________.

Answers: zero; between; outside; values that are not included; values that are included.

Differentiation and Flexibility

• Support: Use a physical number line and ask the learner to count equal distances to both sides of the center. Begin with inequalities whose centers are positive integers.
• Additional scaffold: Provide the sentence frame: “The distance from x to ___ is ___, so x must be between ___ and ___” or “x must be less than ___ or greater than ___.”
• Extension: Have the learner create a real-world situation for |x - 12| ≥ 5, solve it, and explain why the graph shades outward.
• Digital option: Use an online graphing calculator or number-line tool to check the algebraic solution.
• Challenge: Solve an inequality with a coefficient, such as 2|x - 3| ≤ 8. Expected solution: -1 ≤ x ≤ 7.