How to Solve Inequalities Explained — Linear & Quadratic (JAMB/WAEC Mathematics)

An inequality is a mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥, rather than an equals sign, and solving one means finding the full range of values that satisfy that comparison.

This topic is a regular feature in JAMB and WAEC Mathematics, tested through both linear and quadratic inequalities, usually requiring the solution to be expressed as a range or shown on a number line. This lesson walks through both types step by step.

Quick takeaways

• An inequality compares two expressions using <, >, ≤, or ≥, and its solution is usually a range of values, not a single number.
• Solving a linear inequality follows the same steps as solving an equation, EXCEPT that multiplying or dividing both sides by a negative number reverses the inequality sign.
• A quadratic inequality is solved by first finding the roots of the corresponding equation, then testing which regions of the number line satisfy the original inequality.
• Solutions to inequalities are often represented on a number line, using open circles for < or > (excluding the endpoint) and closed circles for ≤ or ≥ (including the endpoint).
• Exam questions frequently test whether students remember to flip the inequality sign when multiplying or dividing by a negative number — this is the single most common error in this topic.

What is an inequality?

An inequality is a mathematical statement that compares two expressions using one of the symbols <, >, ≤, or ≥, rather than an equals sign. While an equation typically has one specific solution (or a small set of solutions), an inequality’s solution is usually a whole range of values that make the comparison true — for example, x > 3 means every value greater than 3 satisfies the inequality, not just one specific number.

How do you solve a linear inequality?

Solving a linear inequality follows the same basic steps as solving a linear equation — isolate the variable using addition, subtraction, multiplication, or division on both sides. For example, to solve 2x + 3 < 11:

• Subtract 3 from both sides: 2x < 8
• Divide both sides by 2: x < 4

What is the sign-flip rule, and why does it matter?

The single most important rule specific to inequalities: whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For example, to solve −2x > 6:

• Divide both sides by −2, and flip the sign: x < −3

Forgetting this rule is the most common mistake in this entire topic, since it’s the one step that doesn’t apply when solving a regular equation, making it easy to overlook under exam pressure.

How do you solve a quadratic inequality?

Solving a quadratic inequality, such as x² − 5x + 6 > 0, requires a different approach from a linear one:

1. Solve the corresponding equation (x² − 5x + 6 = 0) to find the critical values (roots) — in this case, x = 2 and x = 3
2. Plot these critical values on a number line, dividing it into three regions: x < 2, 2 < x < 3, and x > 3
3. Test a value from each region in the original inequality to determine which regions satisfy it
4. State the solution as the range(s) of x where the inequality holds true

For x² − 5x + 6 > 0, testing shows the inequality holds true when x < 2 or x > 3.

How do you represent an inequality’s solution on a number line?

Number lines are commonly used to visually represent an inequality’s solution:

• Open circle (unfilled) — used for < or >, to show the endpoint itself is NOT included in the solution
• Closed circle (filled) — used for ≤ or ≥, to show the endpoint IS included in the solution
• An arrow or shaded region extends from the circle in the direction of all values that satisfy the inequality

Common mistakes students make with inequalities

• Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number
• Treating a quadratic inequality like a quadratic equation, giving only the two critical values instead of the correct range(s)
• Using the wrong type of circle (open vs closed) when representing a solution on a number line
• Forgetting to test which regions actually satisfy a quadratic inequality, and instead guessing based on the sign of the original inequality alone

Frequently asked questions

Q: When do you need to flip the inequality sign? You must flip the inequality sign whenever you multiply or divide both sides of the inequality by a negative number — this is the key difference between solving an inequality and solving a regular equation.

Q: How do you solve a quadratic inequality? First find the roots of the corresponding equation, use these to divide the number line into regions, then test a value from each region in the original inequality to determine which regions satisfy it.

Q: What is the difference between an open circle and a closed circle on a number line? An open circle is used for strict inequalities (< or >) to show the endpoint is excluded, while a closed circle is used for ≤ or ≥ to show the endpoint is included in the solution.

Q: Why does an inequality usually have a range of solutions instead of one specific value? Because an inequality compares two expressions using a “less than” or “greater than” relationship rather than exact equality, so any value satisfying that relationship is a valid solution, resulting in a full range rather than a single number.

Want to strengthen the algebra behind solving equations first? Read our lesson on changing the subject of a formula →

This is one topic from our full JAMB & WAEC Maths course — see everything covered →