Writing Inequalities for Real-World Constraints

6.EE.B.8 — Expressions and Equations

In this lesson:

• Write or from a real-world constraint
• Recognize that inequalities have infinitely many solutions
• Graph solutions on a number line

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Write or for a real-world constraint
2. Recognize that and have infinitely many solutions
3. Graph solutions on a number line: open circle and arrow

What You Already Know: Substitution

Testing :

• : ✓ Yes
• : ✓ Yes
• : ✗ No

Today: write the inequality from a description, then graph all solutions.

Inequalities Describe Constraints or Conditions

An inequality uses or to say a variable must exceed or fall below a threshold.

Three steps to write any inequality:

1. Variable — what's the unknown quantity?
2. Threshold — what's the boundary value?
3. Direction — greater than or less than?

Three-Step Framework: Constraint to Inequality

Write what varies → name it → find the limit → choose > or <

Example 1: Speed Faster Than 65 mph

"You are driving faster than 65 mph."

• Variable: speed →
• Threshold: 65; Direction: faster →

Inequality:

: ✓    : ✗

Example 2: Temperature Colder Than −10°C

"The temperature is colder than −10°C."

• Variable: temperature °C →
• Threshold: −10
• Direction: colder = less than

Inequality:

Test: : ✓   : ✗

Your Turn: More Than $20

"You have more than $20." Let = amount of money.

• Threshold: 20
• "More than" means ___

Write the inequality: ___

Think, then advance for the answer.

More Than $20: Answer and Key Point

Inequality:

• : ✓ Works
• : ✗ No
• : ✗ Equal, not more

"More than $20" does not include exactly $20.

Example 3: Box Shorter Than 6 Inches

"A box must be shorter than 6 inches to fit."

• Variable: height → ; Threshold: 6 →

: ✓   : ✗

and also work — many solutions.

Quick Check: Score More Than 80

"You must score more than 80 points to pass."

Let = score. Write the inequality.

Does your answer exclude exactly 80?

Inequalities Have Infinitely Many Solutions

: every number is a solution —

• No largest solution — for any solution, a bigger one exists
• Boundary (5) divides solutions from non-solutions
• is false — boundary excluded

Testing Many Values for

Between any two true values, infinitely more exist — no complete list is possible.

Strict Inequalities Exclude the Boundary

For or , the boundary is never a solution:

• ? False — nothing is greater than itself
• ? False — nothing is less than itself

We call and strict inequalities because they strictly exclude .

Is the Boundary Value a Solution?

Substitute into :

This is false — ten is not less than ten.

is not a solution. The boundary marks where solutions end.

Transition: Showing All Solutions at Once

We can't list every solution to — there are infinitely many.

But we can show them all at once on a number line.

Two elements communicate the complete picture:

1. Where the solutions start or end (the boundary)
2. Which direction the solutions extend

Number Line: Open Circle and Arrow

To graph or :

• Open circle at — boundary is not included
• Arrow toward solutions:
  • → arrow points right
  • → arrow points left

Graphing : Open Circle, Arrow Right

Open circle at 2 (excluded). Arrow right — all numbers greater than 2 are solutions.

Graphing : Step by Step

Open circle at −3 — −3 is not included. Arrow left — all numbers less than −3 are solutions.

Reading a Graph to Write the Inequality

Given: open circle at 4, arrow pointing right

• Open circle → strict inequality ( or )
• Arrow right → greater than
• Circle at 4 → threshold is 4

Inequality:

Reading graphs is just the reverse process — find the circle, find the arrow direction.

Two Common Graphing Errors to Avoid

Error Analysis: Find and Fix Each Mistake

Graph A: — what's wrong?   Graph B: — what's wrong?

Identify the error before advancing.

Practice: Write and Graph Inequalities

1. "A movie is less than 2 hours." ( = hours) — write and graph
2. "Temperature is above 0°C." ( = temp) — write and graph
3. Open circle at 6, arrow left → inequality?
4. Open circle at −4, arrow right → inequality?

Try all four, then advance.

Practice Answers: Check Your Work

1. — circle at 2, arrow left
2. — circle at 0, arrow right
3. Circle at 6, arrow left →
4. Circle at −4, arrow right →

All four use open circles. Arrows point toward solutions.

Key Takeaways: Inequalities and Number Lines

• Write or : variable, threshold, direction
• Solution set: infinitely many values
• Graph: open circle at + arrow toward solutions

Common errors: closed circle · wrong arrow direction · including

Preview: Relationships Between Two Variables

In 6.EE.C.9, you'll explore how two variables relate — for example, if you buy items at $3 each, total = .

Preview question: Write an inequality for "total cost is less than $30."