Graphing a Single Inequality

Graphing inequalities: the boundary line, solid or dashed, choosing a side with a test point, and three inequalities together to find region R.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580 · 3 real past-paper questions below

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The Boundary Line

An inequality in x and y is a whole region of the grid. Its edge is a straight line: change the sign to = and draw that line.

Worked example

1
$x \leq 3 \Rightarrow x = 3$ — Swap the sign for = to get the edge. Every point on x = 3 has x equal to 3, so it is an upright line through 3 on the x-axis.
2
$y \geq - 1 \Rightarrow y = - 1$ — Same move. Every point on y = −1 is at height −1, so it is a flat line through −1 on the y-axis.
3
$y \leq x + 1 \Rightarrow y = x + 1$ — Same move again. This edge has x in it, so it slopes. You need two points to draw it.
4
$x = 0 : y = 0 + 1 = 1$ — Put an easy x into y = x + 1. With x = 0 you get the point (0, 1).
5
$x = 2 : y = 2 + 1 = 3$ — Pick a second x to get a second point, (2, 3). Join the two points with a ruler and carry the line across the grid.

Rule

A solid line for $\leq$ and $\geq$ : the line is included. A dashed line for $<$ and $>$ : the line is not included.

Which Side?

Pick a test point that is not on the line. If it makes the inequality true, its side is the one you want.

Worked example

1
$y \leq x + 1$ — The inequality. Its edge is the line y = x + 1 from the last card.
2
$(0, 0) : 0 \leq 0 + 1$ — Test (0, 0). It is not on the line, since 0 + 1 is 1, not 0. Put x = 0 and y = 0 in.
3
$0 \leq 1 ✓$ — True. (0, 0) is below the line, so below is the side you want.
4
$(0, 3) : 3 \leq 0 + 1$ — Check with a point from the other side. (0, 3) is above the line.
5
$3 \leq 1 \times$ — False, so above is not wanted. This agrees with the first test.

Common mistake

Using a test point on the line, like (1, 2) for $y \leq x + 1$ . It gives $2 \leq 2$ , which is true whichever side you want, so it tells you nothing. Pick a point clearly off the line.

Three Inequalities Together

The 3-mark question gives three inequalities. Draw all three lines, then find the part of the grid that obeys every one.

Worked example

1
$x \leq 3, y \geq - 1, y \leq x + 1$ — The question. Take one inequality at a time.
2
$x = 3 solid, keep the left$ — Draw x = 3 solid, because ≤ includes the line. Smaller x is to the left, so keep the left side.
3
$y = - 1 solid, keep above$ — Draw y = −1 solid, because ≥ includes it. Bigger y is higher up, so keep the side above.
4
$y = x + 1 solid, keep below$ — Draw y = x + 1 through (0, 1) and (2, 3). The test point (0, 0) gave 0 ≤ 1, true, so keep the side below.
5
$(- 2, - 1), (3, - 1), (3, 4)$ — The part kept by all three is a triangle. Its corners are where each pair of lines cross.
6
$(1, 0) : 1 \leq 3, 0 \geq - 1, 0 \leq 2 ✓$ — Check a point inside the triangle in every inequality. All three are true, so this triangle is R. Write R inside it.

Key idea

Read the instruction. Some questions say shade the region you want; others say shade the parts you do not want. Either way, write R in the region: the label and each correct line earn marks.

Which Method? Read the Question

Draw each line first, then choose each side.

You see	You do
x ≤ a or x ≥ a	upright line x = a
y ≤ b or y ≥ b	flat line y = b
y ≤ mx + c	sloped line, then a test point
Less than, or more than	dashed line
Solve 3x + 1 < 7	See Linear Inequalities

Key idea

For y, greater than means above the line. For x, greater than means to the right.

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