How to Solve Inequalities: Every Type, Step by Step, for 2026

Inequalities look like equations with one symbol swapped, and most of the time they behave that way. But that one swap (= becomes <, ≤, >, or ≥) brings a single rule that wrecks more grades than any other Algebra 1 idea: when you multiply or divide by a negative, you flip the sign. Forget that once and the whole problem is wrong, even though everything else looks fine.

This guide covers every kind of inequality you will see from middle school through Algebra 2, with step-by-step examples, graphing rules, and word problems.

The Five Inequality Symbols and What They Mean

Open circles exclude the endpoint; closed circles include it. Get this right on graphs and you collect easy points.

The One Rule That Trips Everyone

When you multiply or divide both sides of an inequality by a negative number, flip the inequality sign.

The Ultimate Algebra Bundle 2026: From Pre-Algebra to Algebra II

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

Example: −2x > 8.
Divide both sides by −2 (flip the sign):
x < −4.

If you forget to flip, you get x > −4, which is the opposite of the truth. Underline the negative every time you see one.

One-Step Inequalities

Same as one-step equations. Use the inverse operation, flip only when needed.

Solve x + 7 < 15.
Subtract 7 from both sides: x < 8.

Solve x/−4 ≥ 3.
Multiply both sides by −4 (flip): x ≤ −12.

Graph: closed circle on −12, arrow pointing left.

Two-Step Inequalities

Same as two-step equations. Undo addition or subtraction first, then multiplication or division. Flip only when you divide or multiply by a negative.

Solve 3x − 7 ≤ 11.
Add 7: 3x ≤ 18.
Divide by 3: x ≤ 6.

Solve −2x + 5 < 13.
Subtract 5: −2x < 8.
Divide by −2 (flip): x > −4.

Common mistake: students flip when they add or subtract a negative number. You only flip on multiplication or division by a negative. Adding −7 does not require a flip.

Multi-Step Inequalities

Same playbook with more steps: distribute, combine like terms, get the variable on one side, then isolate.

Solve 4(x − 3) > 2x + 6.
Distribute: 4x − 12 > 2x + 6.
Subtract 2x: 2x − 12 > 6.
Add 12: 2x > 18.
Divide by 2: x > 9.

Compound Inequalities

Compound inequalities use the word “and” or “or.”

The Ultimate Middle School Math Bundle: Grades 6-8

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

“And” Inequalities (Intersection)

The variable must satisfy both inequalities at the same time. The solution is the overlap.

Solve −3 < 2x + 1 < 9.
Subtract 1 from all three parts: −4 < 2x < 8.
Divide all three parts by 2: −2 < x < 4.

Graph: open circles on −2 and 4, shaded between.

“Or” Inequalities (Union)

The variable satisfies either inequality. The solution is the combined set.

Solve x < −2 or x > 5.
Graph: open circles on −2 and 5, shaded outward in both directions.

A useful test: “and” gives a bounded segment; “or” gives two rays heading away from each other.

Absolute Value Inequalities

The two-rule trick that solves every absolute value inequality:

• |expression| < a becomes −a < expression < a.
• |expression| > a becomes expression < −a OR expression > a.

Mnemonic: “less than” becomes “and”; “greater than” becomes “or.”

Example 1. Solve |2x − 3| ≤ 5.
Rewrite: −5 ≤ 2x − 3 ≤ 5.
Add 3: −2 ≤ 2x ≤ 8.
Divide by 2: −1 ≤ x ≤ 4.

Example 2. Solve |x + 4| > 7.
Rewrite: x + 4 < −7 OR x + 4 > 7.
Solve each: x < −11 OR x > 3.

Special cases to memorize:
– |expression| < negative number: no solution. Absolute value can never be negative.
– |expression| > negative number: all real numbers. Absolute value is always at least 0.

Inequality Word Problems

The translations that show up most:

Example: “A taxi charges $3 to start and $2 per mile. You have $25. How many miles can you afford?”

The Ultimate Elementary School Math Bundle: Grade 3-5

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

Inequality: 3 + 2m ≤ 25.
Subtract 3: 2m ≤ 22.
Divide by 2: m ≤ 11. You can afford up to 11 miles.

Always re-read the question. “How many” usually asks for the maximum integer that satisfies the inequality, which here is 11.

Graphing Inequalities on a Number Line

Three-step process:
1. Solve the inequality.
2. Draw the endpoint as open (< or >) or closed (≤ or ≥).
3. Shade in the direction of the solution.

For a compound inequality, shade the overlap (“and”) or both pieces (“or”). For an absolute value inequality, follow the rewritten form.

Graphing Inequalities in Two Variables

For y > 2x + 1 (or similar):
1. Graph the boundary line. Solid for ≤ or ≥; dashed for < or >.
2. Pick a test point not on the line (the origin is usually easiest).
3. Plug it in. If true, shade that side. If false, shade the other side.

Example: y > 2x + 1. Test (0, 0): 0 > 1 is false. Shade the side that does not contain (0, 0).

Common Inequality Mistakes

1. Forgetting to flip the sign. Underline negatives before dividing.
2. Open vs. closed circle confusion. ≤ and ≥ are closed; < and > are open.
3. Reversing direction on compound problems. “Less than” means “and”; “greater than” means “or.”
4. Dividing instead of multiplying. Sometimes the cleanest move is multiplying both sides by a denominator. Treat it like an equation.
5. Skipping the answer interpretation. Word problems usually ask for an integer, not a continuous range.

A Quick Cheat Sheet

Frequently Asked Questions

Why do you flip the inequality when multiplying by a negative?
Think of −1 < 3. Multiply both sides by −2: 2 and −6. Is 2 > −6 still true? Yes, but the relationship reversed. Multiplying by a negative reverses every comparison, so the symbol must flip too.

Can an inequality have no solution?
Yes. |x| < −3 has none. Also expressions like 5 < 3 have none after simplification.

Can an inequality have infinite solutions?
Yes. |x| > −3 is true for all real numbers, since absolute value is always non-negative.

Are inequalities tested on the SAT?
Yes, both in algebraic and word-problem form. Compound and absolute value inequalities appear regularly on the Digital SAT.

What is the difference between solving and graphing an inequality?
Solving gives the algebraic answer. Graphing displays it on a number line or coordinate plane. Most tests ask for both.

Closing Thought

Inequalities are equations with one extra rule. Master the flip, memorize the open vs. closed circle convention, and learn the “less than means and, greater than means or” shortcut for absolute value, and the topic stops costing you points.

For more practice, browse our Algebra 1 worksheets and our full Math Topics library. When you are ready for a structured workbook, our Algebra 1 collection covers every inequality type above.

Pre-Algebra for Beginners 2026: The Ultimate Step by Step Guide to Preparing for the Pre-Algebra Test

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.