Absolute Value Inequality Adventure

Materials Needed

• Paper or whiteboard
• Pencil or dry-erase marker
• Number line, drawn or printed
• Two small objects, such as coins or toy figures
• Optional: calculator for checking solutions

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 absolute value inequalities such as |x| < a and |x| > a.
• Graph solutions on a number line and write them using inequality notation.
• Check whether a number is part of a solution set.

Success Criteria

I can solve an absolute value inequality when I can:

1. Identify whether the inequality uses less than or greater than.
2. Choose the correct compound inequality.
3. Graph the solution with an open or closed circle and arrows or shading.
4. Test one value to check my answer.

Introduction: The Distance Challenge — 2 Minutes

Hook: Imagine you are standing at position 0 on a sidewalk. A friend says, “Meet me somewhere less than 3 steps away from me.” Where could you stand?

Use two objects to represent positions on a number line. Explain that both 3 steps to the right and 3 steps to the left are the same distance from zero.

Key idea: Absolute value tells how far a number is from zero, and distance is never negative.

Tell the learner: “Today we will learn how to describe all the numbers that are within a certain distance or farther away from a point.”

Body: I Do, We Do, You Do

I Do: Teacher Modeling — 4 Minutes

Draw a number line and model the inequality:

|x| < 3

1. Read the inequality as: “The distance of x from zero is less than 3.”
2. Numbers within 3 units of zero lie between -3 and 3.
3. Because the inequality is strictly less than, -3 and 3 are not included.
4. Write the compound inequality:

-3 < x < 3

Graph it using open circles at -3 and 3, with the region between them shaded.

Now model a “greater than” example:

|x| > 4

1. Read it as: “The distance of x from zero is greater than 4.”
2. Numbers farther than 4 units from zero are less than -4 or greater than 4.
3. Because the inequality is strictly greater than, -4 and 4 are not included.
4. Write:

x < -4 or x > 4

Graph it using open circles at -4 and 4, shading outward in both directions.

Memory trick:

• Less than means stay inside the distance.
• Greater than means go outside the distance.

We Do: Solve Together — 4 Minutes

Work through each problem together. Ask the learner to explain the meaning before solving.

Problem 1: |x| ≤ 2

1. Ask: “Is the solution inside or outside 2 units from zero?”
2. Expected response: Inside, including the endpoints.
3. Write:

-2 ≤ x ≤ 2

Graph with closed circles at -2 and 2 and shading between them.

Problem 2: |x| ≥ 5

1. Ask: “Is the solution inside or outside 5 units from zero?”
2. Expected response: Outside, including -5 and 5.
3. Write:

x ≤ -5 or x ≥ 5

Graph with closed circles at -5 and 5 and shading outward.

Quick Check

Ask the learner whether each number belongs to the solution of |x| < 3:

• 2: Yes, because its distance from zero is 2.
• -2: Yes, because its distance from zero is 2.
• 3: No, because 3 is not less than 3.
• -4: No, because its distance from zero is 4.

You Do: Solo Number-Line Mission — 3 Minutes

Have the learner solve the following independently. For each problem, the learner should write the solution, graph it, and test one possible value.

1. |x| < 6
2. |x| ≥ 3
3. |x| ≤ 1

Answer key:

• |x| < 6: -6 < x < 6
• |x| ≥ 3: x ≤ -3 or x ≥ 3
• |x| ≤ 1: -1 ≤ x ≤ 1

Real-World Application Challenge — 1 Minute

A delivery robot must stay within 2 meters of its charging station. Let x represent the robot’s position, in meters, relative to the station.

Write an inequality describing the safe zone.

Expected answer: |x| ≤ 2, or equivalently, -2 ≤ x ≤ 2.

Ask: “Why are the endpoints included?” The robot is allowed to be exactly 2 meters away.

Conclusion and Assessment — 1 Minute

Ask the learner to complete this verbal recap:

• “Absolute value represents __________ from zero.”
• “An absolute value inequality with less than usually describes numbers __________.”
• “An absolute value inequality with greater than usually describes numbers __________.”

Expected responses: distance; inside; outside.

Exit Ticket

Solve and graph:

|x| > 2

Expected answer: x < -2 or x > 2, with open circles at -2 and 2 and shading outward.

Differentiation and Extensions

Support

• Use a physical number line and have the learner step left or right to show equal distances.
• Provide the sentence frame: “The distance from zero is ___, so the numbers are inside/outside ___.”
• Begin with inequalities centered at zero before introducing forms such as |x - 2| < 4.

Extension

Challenge the learner to solve:

|x - 2| < 3

Guide them to recognize that this means the distance from 2 is less than 3:

-3 < x - 2 < 3, so -1 < x < 5.

Optional Feedback Prompt

Ask: “What clue helps you decide whether to shade between the endpoints or away from them?”

Look for the learner to explain: less than means inside, while greater than means outside.