Quadratic Inequalities: Solving and Graphing Regions
A quadratic inequality asks for the range of x values that make a quadratic expression positive or negative, rather than for the values that make it zero. The method starts the same way: rearrange so one side is zero, then factorise or use the formula to find the critical values, which are where the curve crosses the x axis. What differs is the last step. Sketching the parabola settles the answer immediately: for an upward parabola the expression is NEGATIVE between the roots and POSITIVE outside them, and for a downward parabola it is the other way round. Answers therefore come in two shapes, a single interval between the roots or a pair of separate regions outside them.
Before you start
Make sure you're comfortable with these topics first:
Method
- Rearrange so that one side is zero, keeping the coefficient of x squared positive where possible, since a negative coefficient means multiplying through by -1 and REVERSING the inequality sign.
- Find the critical values by factorising and setting each bracket to zero, or by using the quadratic formula when it does not factorise.
- Sketch the parabola, marking the two roots on the x axis. The sketch does not need to be accurate, only to show which way up the curve is and where it crosses.
- Read the answer from the sketch: for an upward parabola the curve is below the axis between the roots, so the expression is less than zero there and greater than zero outside them.
- Write the answer as an inequality. Between the roots gives a single statement such as -2 is less than or equal to x is less than or equal to 3, while outside the roots gives two separate statements joined by or.
- Use the correct strictness: strict inequality excludes the roots themselves, while an inequality with an equals part includes them. Test one value from your answer region as a final check.
Worked example
Solve (a) x squared - 5x + 6 is greater than 0, and (b) x squared - x - 6 is less than or equal to 0.
- For (a), factorise: x squared - 5x + 6 = (x - 2)(x - 3), so the critical values are x = 2 and x = 3.
- The coefficient of x squared is positive, so the parabola opens upwards and lies ABOVE the axis outside the roots.
- The solution to (a) is therefore x is less than 2 or x is greater than 3, with strict inequalities because the expression must be strictly greater than zero.
- For (b), factorise: x squared - x - 6 = (x - 3)(x + 2), so the critical values are x = -2 and x = 3.
- Again the parabola opens upwards, so it is BELOW or on the axis between the roots.
- The solution to (b) is -2 is less than or equal to x is less than or equal to 3, including the endpoints because the inequality allows equality. Check with x = 0: 0 - 0 - 6 = -6, which is indeed less than or equal to zero.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Find the critical values of x squared - 7x + 12.Show answer
Answer: It factorises to (x - 3)(x - 4), so the critical values are 3 and 4.
Q2For an upward parabola, where is the expression negative?Show answer
Answer: Between the two roots.
Q3Solve (x - 1)(x - 5) is less than 0.Show answer
Answer: 1 is less than x is less than 5.
Q4Solve (x + 3)(x - 2) is greater than 0.Show answer
Answer: x is less than -3 or x is greater than 2.
Q5What must you do to the inequality sign when multiplying through by -1?Show answer
Answer: Reverse it, so less than becomes greater than and vice versa.
Q6Solve x squared is less than or equal to 16.Show answer
Answer: Rearranged, x squared - 16 is less than or equal to 0, which factorises to (x - 4)(x + 4), so -4 is less than or equal to x is less than or equal to 4.
Q7Why is a sketch the fastest way to finish a quadratic inequality?Show answer
Answer: It shows at a glance which side of the axis the curve is on in each region, so the correct interval can be read off without testing values.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
Solve the inequality 2x squared + 5x - 3 is greater than 0, showing your critical values and your reasoning.
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Solve 12 - x - x squared is greater than or equal to 0. Take particular care with the sign of the x squared term.
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Free printable worksheet
Want more practice on paper? Download the quadratic inequalities: solving and graphing regions worksheet pack - 3 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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