All subjects · Mathematics · WMA11 (P1)

Linear inequalities and regions

Inequalities are solved like equations, with one extra rule. On a graph, an inequality in x and y is a region, not a line.

Solving

Add, subtract, multiply and divide both sides as with equations. If you multiply or divide by a negative number, reverse the inequality sign.

-2x > 6 gives x < -3.

Writing the answer

Two conditions that must both hold can be written together: x > 2 and x < 7 is 2 < x < 7. In set notation: {x : 2 < x < 7}. 'Or' answers can be written as a union, for example {x : x < 1} ∪ {x : x > 4}.

Regions on a graph

Draw the boundary line y = mx + c. Use a dashed line for < or >, because points on the line are not included, and a solid line for ≤ or ≥. y > mx + c is the side where y is larger than on the line. Test a point such as (0, 0) if unsure. Read the question to see whether you shade the wanted region or the unwanted one.

Worked example

Solve 5 - 2x < 3x + 20.

  1. Subtract 5 from both sides: -2x < 3x + 15
  2. Subtract 3x: -5x < 15
  3. Divide by -5 and reverse the sign

Answer: x > -3

Key idea: Reverse the sign when multiplying or dividing by a negative. Dashed boundary for < and >, solid for ≤ and ≥.

Check yourself

  1. Solve 3x - 7 > 11.
    • x < 6
    • x > 4/3
    • x > 6
    • x > 18
    Show answer

    x > 6. 3x > 18, so x > 6.

  2. Solve 4 - 3x ≤ 16.
    • x ≥ -4
    • x ≤ -4
    • x ≥ 4
    • x ≤ 4
    Show answer

    x ≥ -4. -3x ≤ 12. Dividing by -3 reverses the sign: x ≥ -4.

  3. Find the largest integer x that satisfies 2(x + 3) < 17.
    Show answer

    5. 2x + 6 < 17, so x < 5.5. The largest integer is 5.

  4. How many integers satisfy both 2x + 1 > 3 and 3x - 2 ≤ 13?
    Show answer

    4. The first gives x > 1 and the second gives x ≤ 5. The integers are 2, 3, 4 and 5: four of them.

  5. Which is the set notation for 'x > 2 and x < 7'?
    • {x : x > 2} ∪ {x : x < 7}
    • {x : 2 ≤ x ≤ 7}
    • {x : x < 2} ∪ {x : x > 7}
    • {x : 2 < x < 7}
    Show answer

    {x : 2 < x < 7}. Both must hold, so x lies between 2 and 7, not including the ends.

  6. How do you show the region y < 2x + 1 on a graph?
    • Solid line y = 2x + 1; the region under the line
    • Dashed line y = 2x + 1; the region under the line
    • Dashed line y = 2x + 1; the region over the line
    • Solid line y = 2x + 1; the region over the line
    Show answer

    Dashed line y = 2x + 1; the region under the line. The inequality is strict, so the line is dashed. y < 2x + 1 means y is smaller than on the line, so the region is under it.

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Last updated 7 November 2026. Spotted a mistake? Tell us.