Writing Inequalities for Constraints

Lesson: Expressions and Equations

In this lesson:

• Write or to describe real-world limits
• Recognize that inequalities have infinitely many solutions
• Show solution sets on a number line

Learning Objectives for Today's Lesson

You will:

1. Write or for a constraint
2. Test values by substitution to check an inequality
3. Recognize that and have infinitely many solutions
4. Graph solutions using an open circle and arrow

An Elevator Sign Raises a Question

The elevator sign reads: "Maximum Occupancy: 30 people"

If = number of people currently inside:

• Can 25 people ride? Can 31?
• Which values of does the sign allow?

Four Steps to Write Any Inequality

1. Define variable with units — = number of people
2. Identify the threshold — The limit is 30
3. Confirm direction — Below the limit → further left →
4. Write the inequality —

Then test one value from each side.

Elevator Constraint Solved:

Threshold itself: ? No — boundary is never a solution.

Score Must Exceed Eighty to Pass

Constraint: Score greater than 80 to pass.

• Let = score; threshold 80; "greater than" → right →
• Inequality:
• : ? ✓ passes
• : ? ✗ fails
• Boundary: ? No

Colder Means Less Than, Not Greater

Constraint: Temperature fell below C.

• Let = temperature; threshold
• "Below" = colder = smaller = further left →
• Inequality:
• : ? ✓ colder
• : ? ✗ warmer

Your Turn: Flat-Rate Package Weight

Constraint: Package must weigh less than 2.5 lb for flat-rate.

Step 2: Threshold is ___
Step 3: "Less than" → ___ direction → symbol ___
Step 4: Inequality: ___

Test , , and .

The Threshold Always Fails the Test

Testing the threshold always gives false:

• ? No | ? No | ? No

Why? For : substituting gives — always false.
For : substituting gives — always false.

Equations Have One Solution — Inequalities?

How many solutions does have? Think first.

Every Inequality Has Infinitely Many Solutions

For : the solution set has no largest value

• 29 satisfies ; 29.9 satisfies; 29.99 satisfies

For any value you name, subtract 0.001 — still a solution.

Checking Infinitely Many:

Name five solutions to :

Is there a smallest solution? Name one — I'll find a smaller one.

Boundary: Is ? — No. Not a solution.

We Can Show All Solutions at Once

has infinitely many solutions — we cannot list them.

We need a picture that shows the entire solution set at once.

Open Circle and Arrow Explained

• Open circle at threshold → boundary excluded
• Arrow toward solutions → direction from 6.NS.C.7 ordering

Building the Graph for

Step 1: Mark the threshold at 2

Step 2: Test: ? No → open circle at 2

Step 3: Solutions are greater than 2 → further right → arrow points right

Step 4: Check: Is ? Yes ✓ — 5 is in the arrow's direction

Building the Graph for

Step 1: Mark the threshold at

Step 2: Test: ? No → open circle at

Step 3: Solutions are less than → further left → arrow points left

Step 4: Pick : ? Yes ✓ — is to the left? Yes ✓

Read the Diagram, Write the Inequality

For each diagram: find the threshold, note arrow direction, write the inequality.

A: Open circle at 4, arrow right → ___

B: Open circle at 1, arrow left → ___

C: Open circle at , arrow left → ___

Two Common Errors to Diagnose

Two student graphs for and are shown.

Error 1: Closed circle at 4 — is false, so 4 is excluded. Use open circle.

Error 2: Arrow left for — ? No. Arrow must point right.

Key Takeaways and Three Watch-Outs

✓ Four steps: variable, threshold, direction, write the inequality

✓ Threshold is never a solution — is always false

✓ and each have infinitely many solutions

"Colder" = less than — use the number line

Strict inequality → open circle

Arrow direction = values that satisfy — not symbol shape

Inequalities Carry Forward into Algebra

You can now write, interpret, and graph and .

Next: 6.EE.C.9 — inequalities describing two-variable relationships

Later: — same substitution test, same number line logic, more complex expressions