How to Solve Inequalities
Inequalities follow nearly the same steps as equations, with one rule you must not forget. This guide covers linear, compound, quadratic and absolute-value inequalities, and how to write each answer.
The one rule that changes
You may add or subtract the same number on both sides freely. When you multiply or divide by a negative number, reverse the inequality sign. Everything else is as for equations.
Writing the answer
Answers can be written as an inequality (x ≤ −3), on a number line, or in interval notation ((−∞, −3]). Round brackets exclude an endpoint and square brackets include it.
Quadratic inequalities
Move everything to one side, factor, and find the roots. These split the number line into regions. Test one value in each region and keep the regions where the statement is true.
Absolute-value inequalities
|x − a| < k means x is within k of a, so a − k < x < a + k. |x − a| > k means x is further than k from a, which gives two separate pieces. For quick practice, use the inequality solver; for the equation version, see how to solve linear equations.
Worked examples
Example 1: Linear: 5 − 3x ≥ 14
- Subtract 5: −3x ≥ 9.
- Divide by −3 and flip the sign: x ≤ −3.
Final answer: x ≤ −3, or (−∞, −3]
Example 2: Quadratic: x² − x − 6 > 0
- Factor: (x − 3)(x + 2) > 0, with roots −2 and 3.
- Test x = 0: (−3)(2) = −6, which is negative, so the middle region fails.
- Test x = −3 and x = 4: both are positive, so the outer regions work.
Final answer: x < −2 or x > 3
Example 3: Absolute value: |x − 1| < 4
- The distance from 1 is less than 4.
- So −4 < x − 1 < 4. Add 1 to every part.
Final answer: −3 < x < 5, or (−3, 5)
Common mistakes to avoid
- Forgetting to flip the sign when dividing by a negative.
- Treating a quadratic inequality like an equation and stopping at the roots.
- Using brackets of the wrong type in interval notation.
- Writing one inequality where two pieces are needed, as with |x| > k.
- Not testing a value in the final answer.
Frequently asked questions
Why does the sign flip with negatives?
Multiplying by a negative reverses the order of numbers, so 2 < 5 becomes −2 > −5.
What does the circle on a number line mean?
An open circle excludes the number (< or >), a filled circle includes it (≤ or ≥).
How do I solve x² > 9?
Rewrite as x² − 9 > 0, factor to (x − 3)(x + 3) > 0 and test regions: x < −3 or x > 3.
What is interval notation?
A compact way of writing a range, such as [2, 5) for 2 ≤ x < 5.
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