Mathematics › ⚡ Grade 6 › Inequalities: When One Number Isn't the Only Answer

Inequalities: When One Number Isn't the Only Answer

Drag the boundary and flip the relation to see every value that makes the statement true.

Grade 6Middle School

Inequalities: When One Number Isn't the Only Answer — illustration

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The big idea: An equation like x = 5 has exactly one solution. An inequality like x > 5 has infinitely many — every number bigger than 5 works. Substituting a number in checks whether it's a solution; a number line pictures all the solutions at once, with an open or closed circle marking the boundary and a ray showing everything beyond it.

🎯 By the end, you'll be able to

Determine whether a given value makes an inequality true by substituting it in
Write an inequality (x > c or x < c, including ≥ and ≤) to model a real-world constraint
Graph the solution set of a one-variable inequality on a number line
Distinguish an open circle (>, <) from a closed circle (≥, ≤) at the boundary
Recognize that an inequality can have infinitely many solutions, not just one

📎 You should already know

Integers and the number line
Variables and algebraic expressions
Solving one-step equations (6.EE.7)

One solution, or infinitely many?

An equation like x = 5 has exactly one solution: 5. But real life is full of statements that don't pin down one exact number. “You must be at least 12 to ride this coaster” doesn't mean exactly 12 — it means 12, 13, 47, or any age from 12 up.

Statements like this are inequalities. Instead of an equals sign, they use a symbol that compares two amounts: > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to).

Checking a solution by substitution

To check whether a number is a solution to an inequality, substitute it in for the variable and see if the resulting statement is true. Take x > 3: substituting x = 5 gives 5 > 3, which is true, so 5 is a solution. Substituting x = 3 gives 3 > 3, which is false — 3 is not a solution, even though it's the boundary number.

This is the same substitution move you already use to check equations — the only difference is the symbol you're checking against.

Turning words into an inequality

Certain phrases signal which symbol to use. “At least” and “no less than” mean ≥. “At most” and “no more than” mean ≤. “More than” means >, and “fewer than” or “less than” means <.

Example: “A backpack can carry at most 20 pounds” becomes w ≤ 20, where w is the weight in pounds.

🔑 Substitution tells you true or false

Plug the number in, simplify, and ask: is the resulting statement true? If yes, it's a solution. If no, it isn't — no matter how close it is to the boundary.

Inequality	Try x = 3	Solution?
x > 3	3 > 3	False — no
x ≥ 3	3 ≥ 3	True — yes
x < 3	3 < 3	False — no

✨ Infinitely many solutions — one picture

You can't list every solution to x > 3 — there's no largest one. That's exactly why a number line is so useful: instead of listing numbers, you draw a single ray that represents all of them at once, stretching toward infinity.

🎮 Explore an Inequality LIVE

Drag the boundary slider to move a, drag the test-point slider to move x, and press "Change relation" to cycle through >, ≥, <, ≤. Watch the open or closed circle and the shaded ray update, and see whether the test point lands inside the solution set.

⚠️ Open circle vs. closed circle

The most common graphing mistake is picking the wrong circle. Use an open circle for > or < — the boundary number itself is not a solution. Use a closed (filled) circle for ≥ or ≤ — the boundary number is a solution. Then shade a ray in the direction the symbol points: right for > or ≥, left for < or ≤.

📝 Worked example: Write an inequality for: "You must be at least 12 years old to ride this coaster," using a for age. Then describe its graph.

"At least" means the age can be 12 or more, so use ≥: a ≥ 12.
Since ≥ includes the boundary, use a closed circle at 12.
Shade the ray to the right, toward larger ages.

✓ a ≥ 12: closed circle at 12, ray shaded to the right.

📝 Worked example: Is x = 5 a solution to x < 5? Is x = 4?

Substitute x = 5: 5 < 5 is false.
Substitute x = 4: 4 < 5 is true.

✓ x = 5 is not a solution; x = 4 is a solution.

📝 Worked example: A vending machine snack costs at most $3. Write an inequality for the price p, then graph it.

"At most" means the price can be $3 or less, so use ≤: p ≤ 3.
Since ≤ includes the boundary, use a closed circle at 3.
Shade the ray to the left, toward smaller prices.

✓ p ≤ 3: closed circle at 3, ray shaded to the left.

Check your understanding

1. Which value is a solution to x > 7?

Substituting 8 gives 8 > 7, which is true. Substituting 7 gives 7 > 7, which is false, so 7 itself is not a solution.

2. Which inequality represents "you must be at least 18 to vote," using a for age?

"At least 18" means 18 or older, which is a ≥ 18.

3. Which describes the graph of x ≤ 4 on a number line?

≤ includes the boundary, so use a closed circle at 4, and shade left toward smaller numbers.

4. A cooler holds at most 24 cans. Which inequality models the number of cans c?

"At most 24" means 24 or fewer, which is c ≤ 24.

5. Is x = -2 a solution to x < 0?

-2 is less than 0, so substituting gives a true statement: -2 < 0.

✅ Key takeaways

An inequality compares two amounts using >, <, ≥, or ≤, and usually has infinitely many solutions, not just one.
Substitute a value in and check if the resulting statement is true — that's how you test a solution.
Key phrases translate to symbols: 'at least' → ≥, 'at most' → ≤, 'more than' → >, 'fewer than' → <.
On a number line, use an open circle for > or <, a closed circle for ≥ or ≤, then shade a ray in the direction of the solutions.
The boundary number itself is only a solution when the symbol includes 'or equal to.'