Graphical Representation of Inequalities

From an inequality to a region

An equation such as $y=x+2$ describes points on a line. An inequality such as $y\leq x+2$ describes many more points: for each value of $x$, the $y$-coordinate can be any value on or below that line. Its solution set is therefore a region of the coordinate plane, rather than just the line itself.

To draw the boundary, first replace the inequality sign with an equals sign. For $y\leq x+2$, draw $y=x+2$, for instance through $(0,2)$ and . The line is because points on it satisfy the ‘equal to’ part of . With or , use a line: points on the boundary make the two sides equal, so they do not satisfy a strict inequality.

Choosing the correct side

A test point tells you which side of the boundary to show as the solution. For $y\leq x+2$, the origin $(0,0)$ is not on the line. Substituting it gives $0\leq 0+2$, which is true, so the solution region is the side containing the origin. If the test gives a false statement, choose the opposite side. You can often see directly that means ‘below or on the line’, but substitution also works when the inequality is written in a less familiar form. If the boundary passes through the origin, choose another point that is clearly off the line.

Horizontal and vertical boundaries work the same way. The inequality $y>2$ has a dashed horizontal boundary $y=2$ and solutions above it. The inequality $x\leq 3$ has a solid vertical boundary $x=3$ and solutions to its left. Even though only one variable appears, each solution on a coordinate grid is still a point .

Where several regions overlap

For two inequalities to hold at once, a point must satisfy both. Draw each boundary, identify its permitted side and take only the overlap. In the figure, $y\geq 1$ allows points on or above a solid horizontal line, while $y<x+3$ allows points strictly below a dashed diagonal line. The shared region is the solution set.

For example, $(0,2)$ belongs to the shared region: $2\geq1$ and $2<0+3$. A point on $y=1$ can be included if it also satisfies ; a point on the dashed line cannot be included. This is why the style of each boundary still matters where regions overlap.

Reading a graph in reverse

If a question gives you a region and asks for its inequalities, work backwards. Find the equation of each boundary line, using two points on a sloping line if necessary. Then use its solid or dashed style to decide whether equality is allowed. Finally, choose a point inside the required region to decide which inequality sign gives the correct side. Some diagrams shade the required region; others shade the unwanted area and leave the answer clear, so use the question’s labels and wording rather than assuming what shading means.