Teach Me Maths

Inequalities

How to solve linear and quadratic inequalities, handle negative divisors and select intervals or integer solutions.

Use equation methods, with one sign rule

Adding or subtracting the same quantity preserves an inequality. Multiplying or dividing both sides by a negative number reverses the inequality sign. Multiplying by a positive number preserves it.

Strict signs < and > exclude the boundary; \le and \ge include it. Do not multiply by an expression of unknown sign without separating the possible cases.

Worked example 1

Solve 4(2x-3)>5x+6

  1. Expand and collect: 8x-12>5x+6, so 3x>18

  2. Your turn. What is 18\div3?

    18\div3=6

  3. Answer: x>6. Division by positive 3 leaves the sign unchanged.

Worked example 2

Solve 7-3x\le16

  1. Subtract 7: -3x\le9. Divide by -3 and reverse the sign.

  2. Your turn. Give the inequality for x

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    x\ge-3

  3. Answer: x\ge-3. Check the included endpoint: 7-3(-3)=16

Worked example 3

Solve -5<2x+3\le11

  1. Subtract 3 from all three parts: -8<2x\le8

  2. Your turn. Divide all three parts by 2. What is the upper boundary?

    The upper boundary is 4, included.

  3. Answer: -4<x\le4. The lower endpoint is excluded and the upper one is included.

Use roots to split the number line

For a quadratic inequality, move all terms to one side and find the roots. They divide the number line into intervals. Use the curve’s shape or a test value in each interval to decide where the expression is positive or negative.

Roots locate the boundaries; they are not usually the complete inequality answer. Include a root only when equality is allowed.

Explore the sign of a quadratic

Try it: test x=-4, x=0 and x=3. The upward-opening curve is negative between its roots and positive outside them.

Worked example 4

Solve (x-2)(x+5)<0

  1. The roots are -5 and 2. The leading coefficient is positive, so the graph opens upwards.

  2. Your turn. Test the interval between the roots with x=0: calculate (-2)(5)

    The expression is negative between the roots.

  3. Answer: -5<x<2. Both endpoints are excluded because the required inequality is strict.

Worked example 5

Solve 2x^2-5x-3\ge0

  1. Factorise: (2x+1)(x-3)\ge0

  2. Your turn. Find the two roots of the corresponding equation.

    Give every value, separated by commas

    The roots are -\frac12 and 3

  3. The parabola opens upwards, so it is non-negative outside the roots, including them.
  4. Answer: x\le-\frac12 or x\ge3. Use “or” because these are separate intervals.

Worked example 6

Solve -x^2+4x+12>0, then list its positive integer solutions.

  1. The boundary equation has roots -2 and 6, since -x^2+4x+12=-(x+2)(x-6)
  2. The curve opens downwards, so it is positive between the roots: -2<x<6

  3. Your turn. What is the greatest positive integer satisfying this inequality?

    The greatest is 5; 6 is excluded.

  4. Answer: all real solutions satisfy -2<x<6; the positive integer solutions are 1,2,3,4,5

Common mistakes

  • Keeping the sign after multiplying or dividing by a negative. Reverse the inequality sign in either case.
  • Giving roots instead of intervals. Use the roots to split the number line, then test the intervals.
  • Always choosing between the roots. Check the required sign and whether the quadratic opens up or down.
  • Including strict-inequality endpoints. Use open endpoints for < or >; equality can be included for \le or \ge.
  • Confusing “and” with “or”. “And” requires both conditions; “or” allows either condition.

Now try it: Inequalities practice questions

More on this topic: Inequalities worksheet with full solutions