GCSE Maths Inequalities on Graphs Step by Step
Graphical inequalities describe regions of the coordinate plane rather than a single line. GCSE questions often show one or more boundaries and ask which side satisfies an inequality. The most reliable approach is to draw the boundary as an equation, decide whether the line itself is included and test a point away from the line to identify the correct region.
Begin by drawing the boundary line
For y < 2x + 1, first draw y = 2x + 1. Its y-intercept is 1 and its gradient is 2, so it passes through (0,1) and (1,3). The boundary divides the plane into two regions. Draw accurately from a small table of values if the line equation is unfamiliar. The next step is to decide which side to shade, not to guess from the visual slope.
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Know when to use solid and dashed lines
If the inequality is strict, using < or >, points exactly on the boundary do not satisfy it; use a dashed line under the usual GCSE drawing convention. If the inequality includes equality, as in y ≤ 2x + 1, the boundary line is included and is drawn solid. The shading for a strict and non-strict inequality can be identical, but the boundary treatment must differ. Read the symbol closely.
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Test an easy point
Use (0,0) to test y < 2x + 1. Substituting gives 0 < 1, which is true, so shade the side containing the origin. If the boundary passes through (0,0), select another easy point such as (0,1). Write the test inequality explicitly; this checks reasoning and provides an explanation for the chosen region. In many diagrams, this is more dependable than remembering above or below a line.
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Deal with vertical and horizontal boundaries
For x ≥ 2, the boundary is the vertical line x = 2 and the required region is to the right, including the line. For y < −1, the boundary is horizontal at y = −1 and the region is below it, excluding the line. These inequalities do not need an equation in y = mx + c form. Draw each on the correct axis and consider what coordinate is being restricted.
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Combine two inequalities on one graph
Suppose y ≤ x + 2 and y ≥ −x. Draw both boundary lines solid and shade only points satisfying both conditions. A point such as (0,1) satisfies 1 ≤ 2 and 1 ≥ 0, so it lies in the feasible overlap. A point such as (0,−1) fails the second condition and should not lie in the final shaded region. Two separately shaded half-planes must be intersected, not simply combined into their union.
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Common graphical inequality errors
Do not shade the entire plane on both sides of a boundary. Use a test point instead of guessing. Remember that a solid line indicates equality is allowed, and check intersections when more than one condition applies. If a region contains a labelled coordinate, substitute it into every inequality to verify it is feasible. This is a useful final check even when the sketch is not drawn to perfect scale.
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Frequently asked questions
How do I know which side of a line to shade?
Test a point not on the boundary and shade the side that satisfies the inequality.
Do I use a dashed line for ≥?
No. Equality is included with ≥ and ≤, so use a solid boundary line.
Related GCSE revision resources
The existing guides below explore supporting skills. Choose the explanations that match the area where further understanding is required.
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