# Graph Linear Inequalities in Two Variables

## VOCABULARY

Linear inequality in two variables **The result of replacing the 5 sign in a
linear equation with <,≤,> or ≥**

Solution of an inequality in two variables **An ordered pair (x, y) that
produces a true statement when the values are substituted into an inequality in
two variables x and y**

Graph of an inequality in two variables **The set of points that represent all
solutions of the inequality**

Half-plane **Either half of the coordinate plane that is divided by the
boundary line of a linear inequality**

## Example 1 Check solutions of a linear inequality

Tell whether the ordered pair is a solution of 3x - 4y >
9.

a. (2, 0) b. (2, 21)

**Solution**

a.

3x - 4y > 9 Write inequality. 3(** 2 **) - 4(** 0 **)
9 Substitute **2** for x and **0** for
y.** 6** > 9 x Simplify. (2, 0) is not a solution of 3x - 4y > 9.

b.

3x - 4y > 9 Write inequality. 3( 2 ) - 4( -1 )
9 Substitute **2** for x and **-1**
for y. 10 > 9 ✓ Simplify. (2, -1) is a solution of 3x - 4y > 9.

GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES

**Step 1** **Graph** the boundary line. Use a **dashed** line
for < or >, and use a **solid** line for **≤**or ** ≥**.

**Step 2** **Test** a point not on **the boundary line** by
checking whether the ordered pair is a solution of the inequality.

**Step 3** **Shade** the **half-plane** containing the point if
the ordered pair **is** a solution of the inequality. Shade the **
other half-plane** if the ordered pair **is not** a solution.

Example 2 Graph a linear inequality in two variables

Graph the inequality .

**Solution**

**Step 1 Graph** the equation . The
inequality is <, so use a **dashed** line.

**Step 2 Test** (0, 0) in .

**0** <(**0**)+4

**0 **<** 4** ✓

**Step 3 **__Shade__ the half-plane that **contains** (0, 0)
because (0, 0) **is** a solution of the inequality.

## Example 3 Graph a linear inequality in one variable

**Graph the inequality x ≥ 4.**

**Solution**

**Step 1 Graph** the equation x 5 4. The inequality is ** ≥**, so use a
solid line.

**Step 2 Test** (0, 3) in x ** ≥** 4. You only substitute the x-coordinate
because the inequality does not have the variable y . 0 ** ≥** 4 x

**Step 3 Shade** the half-plane that does not contain (0, 3), because (0, 3)
is not a solution of the inequality.

## Guided Practice Graph the inequality.

1. 2y + 4x > 8

2. y < 2