Lesson 8-4: Write and Represent Inequalities

Warm-Up 1

Which symbol means "less than"?

Vocabulary: An inequality is a math sentence that compares two values using one of these symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to). The small pointed end always points to the smaller value.

✓Correct! The symbol < means "less than." The pointed end faces the smaller number.

✗Not quite. The symbol < means "less than." Think of it as an alligator mouth — it opens toward the bigger number. The answer is C.

Warm-Up 2

True or false: The inequality x > 5 means that x can equal 5.

Hint: The symbol > means "strictly greater than." It does not include the number itself. If it did, we would write ≥ (greater than or equal to).

✓Correct! x > 5 means x must be greater than 5, so x cannot equal 5. We use an open circle on the number line for > and <.

✗The symbol > means strictly greater than — 5 itself is not included. The statement is false. We would need ≥ to include 5.

Practice 1

Write the correct inequality symbol for each phrase.

Key phrases:
greater than → > less than → < at least → ≥ at most → ≤ more than → > no more than → ≤

a. "x is greater than 10" → x 10

b. "y is less than or equal to 7" → y 7

c. "n is at least 3" → n 3

Sentence frame: "The phrase '___' tells me to use the symbol ___ because it means ___."

Explain how you chose each symbol:

✓All correct! "Greater than" = >, "less than or equal to" = ≤, and "at least" = ≥.

✗Some symbols need another look. Remember: "at least" means ≥ and "at most" means ≤. Check the key phrases and try again.

Practice 2

A theme park ride requires you to be at least 48 inches tall. Let h represent height in inches. Which inequality represents this situation?

Vocabulary: At least means "that amount or more." If you must be at least 48 inches, you can be 48, 49, 50, and so on — but not 47. That means we need ≥ (greater than or equal to).

Explain your reasoning:

Sentence frame: "The phrase 'at least 48 inches' means h must be ___ or ___. So the inequality is h ___ 48."

✓Correct! "At least 48" means 48 or more, so h ≥ 48. Notice we use ≥ because 48 itself is allowed.

✗"At least" means "that number or more." Since 48 is included, we need ≥, not just >. The answer is B: h ≥ 48.

Practice 3

Sort each inequality into the correct number-line category.

Number line rules:
Open circle = the number is NOT included (< or >).
Closed circle = the number IS included (≤ or ≥).
An open circle means "up to but not equal." A closed (filled) circle means "equal to is okay."

x > 4

y ≥ 2

n < 10

m ≤ 6

t < 1

w ≥ 15

Open Circle (not included)

Closed Circle (included)

✓All correct! Strict inequalities (< and >) use open circles. Inequalities with "or equal to" (≤ and ≥) use closed circles.

✗Some items are in the wrong group. Remember: < and > = open circle (not included). ≤ and ≥ = closed circle (included). Try again!

Practice 4

Which inequality matches this number line?

Reading a number line: Look at two things:
1. Is the circle open or closed?
2. Which direction does the arrow point — left (smaller) or right (larger)?
Below, the circle at 3 is closed and the arrow goes to the right, meaning 3 and everything greater.

Explain how the number line told you the answer:

Sentence frame: "The circle at 3 is ___ (open/closed), so 3 is ___. The arrow points ___, so x is ___ 3. The inequality is ___."

✓Correct! The closed circle means 3 is included, and the arrow points right (greater). So x ≥ 3.

✗Look carefully: the circle at 3 is filled in (closed = included), and the shading goes right (greater). The answer is B: x ≥ 3.

Practice 5

Does the value x = 4 satisfy the inequality x < 4?

How to check: Replace x with 4 in the inequality: 4 < 4. Ask yourself — is 4 less than 4? Remember, < means strictly less than, not equal to.

x < 4 → 4 < 4 ?

Explain your reasoning:

Sentence frame: "When I replace x with 4, I get 4 ___ 4. Since 4 is ___ less than 4, the value ___ satisfy the inequality."

✓Correct! 4 is NOT less than 4 — they are equal. Since < means strictly less than, x = 4 does not satisfy x < 4.

✗4 < 4 is false because 4 is not less than itself. They are equal. So x = 4 does not satisfy x < 4. The answer is No.

Challenge

A movie theater offers a children's ticket for anyone under 13 years old. Let a represent a person's age in years.

Part A: Write an inequality that represents who qualifies for a children's ticket.
Part B: Describe or draw the number line for your inequality. What kind of circle do you use, and which direction does the arrow point?

Think about it: "Under 13" means less than 13 — not equal to 13. Which symbol means less than?
For the number line: since 13 is NOT included, use an open circle at 13. The arrow should point toward numbers that are less than 13 (to the left).

Part A — Your inequality:

Part B — Describe the number line:

Sentence frame: "The inequality is ___. On the number line, I draw an ___ (open/closed) circle at ___ because ___ is/is not included. The arrow points to the ___ because the values are ___ than ___."

🟠 Level 0 — Extra Support

Warm-Up

Practice

Challenge