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MathidiaMathematics Academy
Advanced equations and inequalities
Lesson 25 of 75
Review

Inequalities as regions

Objective. Interpret simple linear inequalities graphically.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Inequalities as regions represent the set of points satisfying a linear inequality as a shaded area on a coordinate grid, bounded by the line formed when the inequality is replaced by an equals sign. To draw the region for y < 2x + 1, first plot the line y = 2x + 1 using a dashed line because the inequality is strict, then shade the side of the line containing points that satisfy the inequality, often checked using the origin as a test point. If the inequality includes 'or equal to' (≤ or ≥), the boundary line is drawn solid to show it is included in the region. Multiple inequalities combined on one grid define an overlapping region satisfying all conditions at once.

Worked example

Describe how to draw the region that represents the inequality y ≥ x − 2 on a coordinate grid.

2
Coordinate grid with the line 2
  1. 1Draw the boundary line y = x − 2 using a solid line, since the inequality includes 'equal to'.
  2. 2Choose a test point not on the line, such as (0,0).
  3. 3Substitute into the inequality: 0 ≥ 0 − 2, which simplifies to 0 ≥ −2, true.
  4. 4Since the test point satisfies the inequality, shade the region containing (0,0).

Answer: Shade above the solid line y = x − 2, the side containing the point (0,0)

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Common mistake. Pupils often shade the wrong side of the line because they substitute a test point incorrectly, such as checking (0,0) against y < 2x + 1 without carefully comparing both sides. Always substitute the test point into the original inequality and check if the true statement matches the shaded side.

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