Graphing Systems of Inequalities

Key Questions

• A linear inequality is in the form:

$a x + b y > c$ or ax + by < c.

First, graph the line ax + by - c = 0 by the 2 y-intercepts and x-intercepts.

Then find the area (above or below the line) that make the inequality true. You may use the origin O (0, 0) as test point

Example. Solve: 2x + 3y > -4

First, graph the line 2x + 3y + 4 = 0 by the 2 x and y intercepts.
Make x = 0 -> 3y = -4 -> y = -4/3
Make y = 0 -> 2x = -4 -> x = -2
Solve the inequality: f(x) = 2x + 3y + 4 > 0 by using origin O (0, 0) as test point. We get 4 > 0, It is true, then the area above the line is the solution set.
graph{2x + 3y + 4 = 0 [-10, 10, -5, 5]}

• By linear inequality in one variable, do you mean something like

$x \le 5$

or

$y > 2$

If yes, just draw your line and shade the entire area that satisfies your inequality

For example, in $x \le 5$,
we should draw a vertical line at $x = 5$ and shade the entire area to its left (since $\le$)

For y > 2, we should draw a broken horizontal line (since $>$, not $\ge$) at y = 2 and shade the entire area above it