An inequality is like an equation, but instead of saying "two things are equal," it says "one thing is bigger" or "one thing is smaller." Equations have a single answer; inequalities have an entire range of answers. Most real-world problems are inequalities in disguise — "I have at most $50 to spend," "the package must weigh under 5 kg," "we need at least 30 students for the trip." This guide covers what inequalities are, how to solve them, the one weird rule that flips the symbol, and how to draw the answer on a number line.
An inequality uses one of four symbols to compare two expressions:
| Symbol | Meaning | Example |
|---|---|---|
| < | Less than (strictly smaller) | x < 5 means x is less than 5 |
| > | Greater than (strictly bigger) | x > 3 means x is greater than 3 |
| ≤ | Less than or equal to | x ≤ 7 means x can be 7 or any smaller number |
| ≥ | Greater than or equal to | x ≥ −2 means x can be −2 or any bigger number |
An inequality's solution is not a single number — it is an entire set of numbers. The answer to x + 4 < 10 is not "x = 6"; it is "every number smaller than 6."
Solving an inequality uses the same inverse operations as solving an equation: undo the addition or subtraction first, then undo the multiplication or division. Whatever you do to one side, do to the other. There is exactly one extra rule to remember.
Whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. < becomes >, ≤ becomes ≥, and vice versa. This rule does not apply when multiplying or dividing by a positive number — only by a negative.
Inequalities are typically visualized on a number line. Two visual conventions tell the reader exactly which numbers belong to the solution:
For example, the graph of x > 4 is an open circle at 4 with an arrow shading everything to the right. The graph of x ≥ −2 is a closed circle at −2 with an arrow shading everything to the right.
A compound inequality combines two inequalities into one statement. Two main flavors come up.
Sometimes written as a single sandwiched inequality: 3 < x < 8. This means x is greater than 3 and less than 8 — a range. The solution is the overlap of the two conditions. On a number line, it shows as a segment between two circles.
Written with the word "or": x < 2 or x > 6. The solution is anything that satisfies either condition. On a number line, it shows as two separate arrows pointing outward, with a gap in the middle.
The single biggest cause of wrong answers on inequality problems. Every time you multiply or divide by a negative number, the inequality symbol must reverse. Write it on a sticky note until it is second nature.
Adding a negative is not the same as multiplying by a negative. If you subtract from both sides — even subtracting a positive number — the sign stays the same. Only multiplication/division by a negative triggers the flip.
Strict inequalities (<, >) use open circles. Inclusive inequalities (≤, ≥) use closed circles. Mixing these up loses easy test points.
In 4 < 2x + 6 < 14, every operation must be done to all three parts of the chain — not just the middle. Subtract 6 from all three: −2 < 2x < 8. Divide all three by 2: −1 < x < 4.
Most decisions in the real world involve inequalities, not equations. A factory wants at least 1,000 widgets per day. A doctor sets the medication dose not to exceed 50 mg. A budget allows spending up to $200. None of these have a single "correct" answer — they describe ranges of acceptable values. Linear programming, used in supply chain optimization and economics, is built entirely on systems of inequalities.
For students, mastering inequalities prepares you for systems of inequalities in Algebra 1, optimization problems in pre-calculus, and feasible regions in linear programming. The skills also appear directly in calculus, statistics (confidence intervals), and computer science (algorithm bounds).
After solving an inequality, pick any number from your solution set and plug it back into the original. If the inequality is true, your answer is correct. If it is false, you have an error — most likely a missed sign flip. Testing a single point catches almost every mistake.
Basic comparison symbols (< and >) appear in early elementary school. Solving simple one-step inequalities begins around 6th grade. Two-step and compound inequalities are typically covered in 7th and 8th grade pre-algebra, and Algebra 1 revisits them more rigorously with graphing.
Try a concrete example. We know 3 < 5 is true. Multiply both sides by −1: we get −3 and −5. Now ask, is −3 < −5? No — −3 is actually greater than −5 because it sits to the right on the number line. So the inequality has reversed: −3 > −5. Multiplying by a negative reflects the numbers across zero, which reverses their order.
Look at the connecting symbol. An equation uses an equal sign (=). An inequality uses one of <, >, ≤, or ≥. In word problems, look for phrases like "at least," "no more than," "exceeds," "fewer than," "maximum," and "minimum" — they all signal inequalities.
Almost — you cannot just plug in your solution because there are infinitely many solutions. Instead, pick any one number from your solution set, substitute it into the original inequality, and verify the result is true. If you also want extra confidence, pick a number outside the solution and verify the inequality is false there.
"And" means both conditions must be true at the same time — the solution is the overlap (a single segment on the number line). "Or" means at least one condition must be true — the solution is the combined region (often two separate pieces). 2 < x < 5 is an "and"; x < 2 or x > 5 is an "or."
Once you have linear inequalities mastered, the natural next steps are systems of inequalities (multiple inequalities considered together, drawn as shaded regions in the coordinate plane), absolute value inequalities, and eventually quadratic and polynomial inequalities in Algebra 2 and pre-calculus.
Ready to practice inequalities?
Practice Inequalities Now →