How to Solve a Quadratic Inequality

Solving a quadratic inequality means finding every value of $x$ that makes a quadratic expression greater than, less than, or equal to zero. Unlike a quadratic equation, which has at most two answers, an inequality has a whole range of answers, usually an interval between the roots, or the two outer regions beyond them.

The method turns the inequality into a picture: find where the parabola meets the $x$-axis, then read off where the curve sits above the axis (positive) or below it (negative). This answers questions such as "solve $x^2-x-6>0$", "find the range of values of $x$ satisfying $x^2+2x-8\le 0$", or word problems that require a quantity to stay positive.

It is a core Form 4 Quadratic Functions skill, and it depends directly on factorising and on knowing which way a parabola opens.

Because the answer is a range, the way you write it matters as much as the numbers. A clear sketch keeps the direction of each inequality sign honest and stops the two cases, inside the roots versus outside the roots, from being mixed up.

Reach for this method whenever a question contains a quadratic together with an inequality sign, $>$, $<$, $\ge$ or $\le$, and asks for the values of $x$ that satisfy it. Wording such as "solve the inequality", "find the range of values of $x$", or "for what values of $x$ is the expression positive" is the direct cue.

It is also the final step in problems where an earlier part produces a quadratic condition, for instance after using the discriminant to find a range of $k$, or when a length or area must be positive. Because the solution is a set of values rather than a single number, watch for the plural "values".

Quadratic-inequality questions appear in both Paper 1 and Paper 2, often as a short but easily-lost few marks.

Follow this order so the direction never goes wrong:

1. 1

   Make one side zero

   Rearrange the inequality so all terms are on one side and the other side is 0, keeping the coefficient of $x^2$ positive if you can.

2. 2

   Find the critical values

   Solve the matching equation (set the expression equal to zero) by factorising to get the roots. These are the boundary points.

3. 3

   Sketch the parabola

   Draw a quick parabola opening upward, crossing the $x$-axis at the two roots. This shows where the curve is above or below the axis.

4. 4

   Choose the correct region

   For $>0$ or $\ge 0$, take where the curve is above the axis (outside the roots); for $<0$ or $\le 0$, take where it is below (between the roots).

5. 5

   Write the solution

   State the range using the correct signs. Use $\le$ or $\ge$ (roots included) for a non-strict inequality, and $<$ or $>$ (roots excluded) for a strict one.

Try this yourself first, then check each line against the solution.

The test-point check is the fastest safeguard. Pick any easy value inside your proposed range, substitute it, and confirm the inequality really holds.

If it does not, you have taken the wrong region and can fix it before writing the final line.

• Choosing the wrong region, taking "between the roots" when the question wants "outside", or the reverse. A quick sketch prevents this every time.
• Not moving everything to one side first, so the comparison is not really with zero.
• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
• Mixing up strict and non-strict signs, so the endpoints are wrongly included or excluded. Use $\le,\ge$ to include roots and $<,>$ to exclude them.
• Writing the answer as $-4\le x\ge 2$ or joining two separate regions with "and"; a single interval uses one continuous chain, and two outer regions use "or".

The step that loses marks is almost always choosing the region, deciding whether the answer is between the roots or outside them. In a one-to-one lesson our teacher has you draw the little parabola every time, so the choice is read off the picture rather than guessed.

Because our teachers are experienced, you get someone who spots whether your slip is the region, a reversed sign, or an included endpoint, and drills just that. Lessons are online and taught in English, while SPM papers are set in both Malay and English, so we make sure you can express the range clearly in either.

We also build the test-point habit, substitute one easy value to confirm the region, so a careless choice never survives to the answer line.