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Method

Graphing a Linear Inequality

Graphing a linear inequality is similar to graphing a linear equation in slope-intercept form. However, instead of a line, the graph of a linear inequality is a region of the coordinate plane. Consider the following linear inequality.

9 x + 3 y \leq 6

To draw its solution set, the procedure begins by writing the inequality in slope-intercept form. This way, the equation for the boundary line can be obtained. Then, this boundary line is graphed in the coordinate plane. Finally, the region that contains the solutions is shaded.

Write the Inequality in Slope-Intercept Form

To find the boundary line of the region, start by writing the inequality similar to a linear equation in slope-intercept form. This is done solving the inequality for one of the variables, most commonly

y .

9 x + 3 y \leq 6

Write in slope-intercept form

SubIneq

$LHS - 9 x \leq RHS - 9 x$

3 y \leq - 9 x + 6

DivIneq

$LHS / 3 \leq RHS / 3$

y \leq \frac{- 9 x + 6}{3}

WriteSumFrac

Write as a sum of fractions

y \leq \frac{- 9 x}{3} + \frac{6}{3}

MoveNegNumToFrac

Put minus sign in front of fraction

y \leq - \frac{9 x}{3} + \frac{6}{3}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

y \leq - \frac{9}{3} x + \frac{6}{3}

CalcQuot

Calculate quotient

y \leq - 3 x + 2

Then, the inequality in slope-intercept form can be written as follows.

y \leq - 3 x + 2

Graph the Boundary Line

The boundary line of the inequality is obtained by replacing the inequality symbol with an equals sign.

Inequality y \leq - 3 x + 2 Boundary Line y = - 3 x + 2

If the inequality is strict, the points on the line are not solutions to the inequality. Therefore, the line is dashed. Conversely, if the inequality is not strict, the points on the line are solutions to the inequality. In this case, the line is solid.

Symbol	Meaning	Type	Boundary Line
$<$	Less than	Strict	Dashed
$>$	Greater than	Strict	Dashed
$\leq$	Less than or equal to	Non-strict	Solid
$\geq$	Greater than or equal to	Non-strict	Solid

Therefore, in the given example, the line is solid. The boundary line can be graphed using $y -$ intercept and slope.

Test a Point

The region of the coordinate plane either to the left or to the right of the boundary line contains the solution set. To determine the correct region, substitute an arbitrary test point not on the boundary line into the inequality. It is common to use

(0, 0) .

y \leq - 3 x + 2

SubstituteII

$x = 0$ , $y = 0$

0 \leq ? - 3 (0) + 2

Evaluate right-hand side

ZeroPropMult

Zero Property of Multiplication

0 \leq ? 0 + 2

IdPropAdd

Identity Property of Addition

0 \leq 2 ✓

Since

0 \leq 2

is a true statement, the point

(0, 0)

is a solution to the inequality.

Shade the Correct Region

If the test point is a solution to the inequality, the region that contains it must be shaded. Otherwise, the opposite region must be shaded.

In this case, the test point $(0, 0)$ is a solution to the inequality. The region containing $(0, 0)$ lies to the left of the boundary line. This is the region that must be shaded.

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