What Is An Inequality In Math? Solve And Graph It

The short version

An inequality is a math sentence that says two values aren’t equal, using a symbol like <, >, ≤, ≥ or ≠.
x > 5 means “x is greater than 5.” It has endless answers (6, 7.5, 100 and so on), which is why we graph it on a number line.
You solve an inequality almost exactly like an equation. The one extra rule: if you multiply or divide both sides by a negative number, flip the symbol.
On a graph, an open circle means the endpoint isn’t included (< or >). A filled circle means it is (≤ or ≥).

An inequality in math is a statement that compares two values and says one is bigger than, smaller than, or not equal to the other. Instead of an equals sign, it uses a symbol like < (less than) or ≥ (greater than or equal to). So 3 < 8 is an inequality, and so is x + 2 ≥ 10.

The meaning of inequality in mathematics comes down to this: an equation pins down one exact answer, while an inequality describes a whole range of answers that work. That’s the idea kids need most, and it’s the one most explanations rush past. Below we cover the symbols, how to solve and graph an inequality, the sign flip rule and why it works, word problems, and practice with an answer key.

Inequality symbols and how to read them

Symbol	Read as	Example	Words that often signal it
<	is less than	2 < 9	fewer than, below, under
>	is greater than	9 > 2	more than, above, over
≤	is less than or equal to	x ≤ 4	at most, no more than, maximum
≥	is greater than or equal to	x ≥ 4	at least, no less than, minimum
≠	is not equal to	x ≠ 0	can’t be, isn’t

A reliable way to read < and >: the open side faces the bigger number and the point faces the smaller one. Younger kids often learn this as a hungry alligator that eats the bigger number. That’s fine for comparing 7 and 3, but by middle school it’s better to just read left to right. In x < 6, the thing on the left (x) is less than the thing on the right (6).

The line under ≤ and ≥ is half of an equals sign. It’s telling you “or equal to,” so the endpoint counts.

Inequality vs equation

The equation x + 3 = 8 has one answer: x = 5. The inequality x + 3 > 8 has infinitely many. Any number bigger than 5 works, including 5.1, 6, 20 and a million. The number 5 itself doesn’t work, because 5 + 3 = 8, and 8 isn’t greater than 8.

That’s why the answer to an inequality is written as another inequality (x > 5), and why graphing matters. You can’t list every answer, so you draw them.

How to graph an inequality on a number line

Find the boundary number and mark it on the number line.
Draw an open circle for < or > (the boundary isn’t a solution). Draw a filled circle for ≤ or ≥ (it is).
Shade toward the solutions. With x on the left, “less than” shades left and “greater than” shades right.

Example: graph x ≤ -2.
Boundary: -2
≤ includes the endpoint, so fill in the circle at -2
Less than means shade to the left, toward -3, -4, -5 and beyond.

A quick check: pick any number in the shaded part and plug it in. -5 ≤ -2 is true, so the shading is on the correct side.

How to solve an inequality

Use the same moves you’d use on an equation. Add, subtract, multiply or divide both sides by the same amount to get the variable alone.

Example 1: x + 7 > 12
Subtract 7 from both sides: x > 12 – 7
x > 5

Example 2: 3x – 4 ≤ 11
Add 4 to both sides: 3x ≤ 15
Divide both sides by 3: x ≤ 5
Check with x = 5: 3(5) – 4 = 11, and 11 ≤ 11 is true.

The flip rule: multiplying or dividing by a negative

This is the one step that’s different from equations. When you multiply or divide both sides by a negative number, reverse the direction of the symbol.

Example 3: -2x + 5 < 13
Subtract 5 from both sides: -2x < 8
Divide both sides by -2 and flip the symbol: x > -4
Check with x = 0: -2(0) + 5 = 5, and 5 < 13 is true. Zero is greater than -4, so it should work, and it does.

Why the sign flips

Start with something obviously true: 2 < 3. Multiply both sides by -1 and you get -2 and -3. On a number line, -2 is to the right of -3, so -2 > -3. Multiplying by a negative mirrors every number across zero, and that mirror swaps which one is bigger. The symbol has to flip to keep the statement true.

Adding or subtracting a negative doesn’t cause a flip. Only multiplying or dividing does.

Compound inequalities and word problems

A compound inequality puts two conditions together. -1 < x ≤ 4 means x is greater than -1 and less than or equal to 4. On a number line, that’s an open circle at -1, a filled circle at 4, and shading between them.

To solve one, do the same step to all three parts.

Example 4: 1 ≤ x + 3 < 9
Subtract 3 from all three parts: 1 – 3 ≤ x < 9 – 3
-2 ≤ x < 6

Example 5: a word problem. You have $30 to spend on T-shirts that cost $8 each, plus $5 shipping. How many can you buy?
Let n be the number of shirts: 8n + 5 ≤ 30
Subtract 5: 8n ≤ 25
Divide by 8: n ≤ 3.125
You can’t buy part of a shirt, so the most you can buy is 3.

That last step matters. The math says n ≤ 3.125, but the real answer has to make sense for the situation. Rounding up to 4 would put you over budget (8 × 4 + 5 = 37).

Common mistakes with inequalities

Forgetting to flip. In -3x > 12, dividing by -3 gives x < -4. A kid who writes x > -4 skipped the flip. Plugging in a test number catches this every time.
Flipping when you shouldn’t. Some kids flip any time they see a negative, like in x – 4 > -10. Subtracting or adding never flips the sign. The answer is x > -6.
Open vs filled circle mixups. x > 3 gets an open circle, since 3 isn’t a solution. x ≥ 3 gets a filled one.
Reversing “at least” and “at most.” “At least 48 inches tall” means h ≥ 48. Kids see “least” and reach for <.
Reading the symbol backward when the variable is on the right. 7 > x means x is less than 7. Rewriting it as x < 7 before graphing helps.

Practice problems

True or false: 7 > -9.
Write as an inequality: “x is at most 10.”
Solve: x – 6 ≥ 2
Solve: 4x < 28
Solve: -3x ≥ 12
Solve: 5 – 2x > 11
Pizzas cost $12 each and delivery is $6. You have $60. Write and solve an inequality to find the most pizzas you can order.
Solve and describe the graph: 2 ≤ x + 3 < 7

Answers

True. Any positive number is greater than any negative number.
x ≤ 10
x ≥ 8 (add 6 to both sides).
x < 7 (divide both sides by 4; it’s positive, so no flip).
x ≤ -4 (divide by -3 and flip). Check x = -5: -3(-5) = 15, and 15 ≥ 12 is true.
x < -3. Subtract 5: -2x > 6. Divide by -2 and flip: x < -3. Check x = -4: 5 – 2(-4) = 13, and 13 > 11 is true.
12p + 6 ≤ 60, so 12p ≤ 54 and p ≤ 4.5. The most you can order is 4 pizzas ($54 total).
-1 ≤ x < 4. Filled circle at -1, open circle at 4, shading between them.

What it means if your child is stuck on inequalities

Solving an inequality is really solving an equation with one extra rule, so a kid who can’t do it usually has trouble one layer down. The common culprits are negative numbers (where does -4 sit compared to -1?), inverse operations (why do we subtract 7 to undo +7?) and plain comfort with the number line. If equations with one variable are already shaky, inequalities will feel twice as hard.

We see the same chain in most early algebra trouble, and we lay it out in Struggling With Algebra? The Real Problem Is Usually Fourth Grade Fractions. If you’re not sure whether this is a rough week or a real gap, Is My Child Behind In Math? 12 Signs By Grade Level gives you concrete things to look for, and our guide to math assessment for kids explains how to pin it down.

Frequently asked questions

What is the meaning of inequality in mathematics?

It’s a statement that two expressions aren’t equal, written with <, >, ≤, ≥ or ≠. It usually describes a range of values instead of a single answer.

When do you flip the inequality sign?

Only when you multiply or divide both sides by a negative number. You also reverse it if you swap the two sides, so 5 > x becomes x < 5.

What’s the difference between < and ≤?

< means strictly less than, so the boundary number doesn’t count. ≤ means less than or equal to, so it does. On a graph, that’s the difference between an open and a filled circle.

What grade do kids learn inequalities?

Kids compare numbers with < and > in the early grades. Writing and graphing simple inequalities with a variable usually starts around sixth grade, and solving multi-step ones, including the flip rule, comes in seventh grade and Algebra 1.

How do you check the answer to an inequality?

Pick a number from your solution and plug it into the original problem. If the statement is true, and a number outside your solution makes it false, you’re right.

Find the actual gap

More inequality worksheets won’t help much if the real trouble is negative numbers or two-step equations. Math Anon begins with a short adaptive assessment that finds exactly which skills are missing and sends you a written report, so you know what to fix first.

Then your child does about 20 minutes a day of practice built around those specific gaps, and you get weekly progress reports. The assessment costs $49 and is credited back in full on the first month. Start the assessment, or look over pricing first.