Graphing inequalities | Geometry | Quantitative reasoning

An equation with $x$ and $y$ (like $y = m x + b$ ) graphs as a line on the coordinate plane. If you replace the equal sign with an inequality, the boundary line stays the same, but the meaning changes.

With an inequality, the solutions aren’t just the points on the line. Instead, the solutions form a region on one side of the line - either above it or below it. For example, this inequality is written in the same style as $y = m x + b$ , but its solutions are all the points above the line rather than on the line.

graphing inequalities 01

It’s usually straightforward to decide which side of the line to shade.

First, rewrite the inequality in a form like $y (inequality) m x + b$ .
Then use the inequality symbol:
- If the inequality is $>$ or $\geq$ , shade above the line.
- If the inequality is $<$ or $\leq$ , shade below the line.

Sidenote

Dotted line vs. solid line

This is relatively unimportant for the GRE, but a dotted line represents $>$ or $<$ while a solid line represents $\geq$ or $\leq$ . When the inequality includes “or equal to” then the coordinate pairs could also exist on the line.

Try to match each inequality with a graph drawn below.

$y > 2$

$y > x$

$y \leq - 2 x - 2$

$y \geq - 2 x - 2$

graphing inequalities 02

(spoiler)

Answers:

graphing inequalities 03

Now try this GRE-level practice problem that uses the idea of graphing inequalities.

In which quadrants does the intersection of both inequalities exist? (Select all that apply.)

$y \leq 3 x + 2$
$y \leq x - 4$

A. Quadrant I
B. Quadrant II
C. Quadrant III
D. Quadrant IV

(spoiler)

Answer: A, C, D

Explanation:

Start by graphing both inequalities.

graphing inequalities 04

The first inequality’s shaded region appears in all four quadrants (if you look closely, you can see that it barely touches quadrant II). The second inequality’s shaded region appears only in quadrants I, III, and IV.

The intersection is the region that satisfies both inequalities, so it can only occur in quadrants that both shaded regions share. Therefore, the intersection exists in quadrants I, III, and IV.

For context, the union would represent all quadrants where either inequality has solutions. If the question asked for the union instead of the intersection, all four answer choices would be correct.