NASA Mission Control: Solving Absolute Value Equations and Inequalities

Materials Needed

• Pencil and paper or a whiteboard
• Number line, ruler, or NASA “mission timeline” strip
• Calculator, optional
• Timer, optional

Learning Target

I can solve simple absolute value equations and inequalities and represent the solutions algebraically or graphically.

Success Criteria

By the end of the lesson, I can:

• Explain that absolute value represents distance from zero.
• Solve an equation such as |x - 3| = 5.
• Decide whether an absolute value inequality uses “between” or “outside.”
• Graph my solution on a number line.
• Use an absolute value inequality to describe a NASA-related situation.

Introduction and Bell Work: Mission Launch Review — 2 Minutes

Hook: NASA engineers must keep spacecraft measurements within safe limits. Today, you are the mission-control mathematician. Your job is to determine which values are safe and which values are not.

Bell Work: Complete these quick review questions.

1. What is |-7|?
2. What two numbers are 4 units away from 2?
3. Write an inequality that means “x is less than 6.”

Transition: Absolute value tells us distance, and distance is never negative. We will use that idea to solve NASA mission problems.

Body: Gradual Release of Responsibility

I Do: Teacher Modeling — 4 Minutes

1. Absolute Value Equations

Consider the equation:

|x - 4| = 3

This means that x is 3 units away from 4. There are two possible locations:

• x - 4 = 3, so x = 7
• x - 4 = -3, so x = 1

Solution: x = 1 or x = 7

Rule: If |expression| = number, write two equations: one positive and one negative.

2. Absolute Value Inequalities

Consider:

|x - 2| < 4

This means x is less than 4 units from 2. It must stay between 2 − 4 and 2 + 4:

-4 < x - 2 < 4

-2 < x < 6

Solution: -2 < x < 6

On a number line, use open circles at -2 and 6, and shade between them.

Now consider:

|x - 5| > 2

This means x is more than 2 units away from 5. It must be outside the interval:

x - 5 < -2 or x - 5 > 2

x < 3 or x > 7

Memory tip:

• < or ≤ means between.
• > or ≥ means outside.

We Do: Mission-Control Practice — 4 Minutes

Work through each problem together. Explain each step aloud, draw a number line, and check whether the answer makes sense.

Problem 1: Rocket Navigation

A rocket’s horizontal position is modeled by |x - 6| = 4. What are the two possible positions?

Guided steps:

1. Write two equations: x - 6 = 4 and x - 6 = -4.
2. Solve each equation.
3. State both possible positions.

Answer: x = 10 or x = 2

Problem 2: Safe Temperature Range

A spacecraft instrument works safely when its temperature is within 5 degrees of 20°C. Write and solve an inequality.

Guided steps:

1. Represent “within 5 degrees of 20” as |t - 20| ≤ 5.
2. Rewrite as -5 ≤ t - 20 ≤ 5.
3. Add 20 to all three parts.
4. Graph the answer using closed circles because the inequality includes equality.

Answer: 15 ≤ t ≤ 25

Quick Check

Ask: Is 24°C safe? Is 27°C safe? Explain why.

Expected response: 24°C is safe because it is within 5 degrees of 20. 27°C is not safe because it is 7 degrees away.

You Do: Independent NASA Challenge — 4 Minutes

Choose either the Orbit Challenge or the Mission Challenge. Show your algebraic work and graph each solution on a number line.

Option A: Orbit Challenge

1. Solve: |x + 2| = 6.
2. Solve: |x - 1| < 3.

Option B: Mission Challenge

NASA allows a satellite to be no more than 8 kilometers from its planned position of 50 kilometers.

1. Write an absolute value inequality.
2. Solve the inequality.
3. Describe the answer in words.
4. Graph the solution.

Expected answer for Option B:

|x - 50| ≤ 8

42 ≤ x ≤ 58

The satellite may be from 42 kilometers through 58 kilometers, including both endpoints.

Conclusion and Mission Debrief — 1 Minute

Complete the following exit ticket without looking back at the examples.

1. Solve |x - 4| = 2.
2. Solve |x + 1| ≤ 3.
3. Which type of absolute value inequality describes values between two endpoints: less than or greater than?

Student recap: Complete this sentence: “Absolute value inequalities help me describe __________, and when the inequality is less than, I shade __________.”

Suggested response: “distance from a number; between the endpoints.”

Assessment

Formative Assessment

• Review bell work responses.
• Listen for explanations during the We Do practice.
• Check whether the learner chooses “between” or “outside” correctly.
• Use the quick check to test understanding of a real-world inequality.

Summative Assessment

The learner demonstrates mastery by completing the exit ticket with at least 2 out of 3 correct and by including a correct graph for at least one inequality.

Differentiation and Support

• Scaffold: Use a number line and physically count the distance from the center number. Provide the sentence frame: “The distance from x to ___ is ___.”
• Extra support: Begin with equations before introducing inequalities. Highlight the center number and distance in different colors.
• Verbal support: Have the learner explain “between” and “outside” using hand motions or by standing on a floor number line.
• Extension: Create a NASA scenario involving a launch angle, temperature, altitude, or satellite position. Write an absolute value inequality, solve it, and graph the solution.
• Digital option: Use an online number-line tool or graphing calculator to verify the solution.