Method

Writing a Linear Inequality

Writing a linear inequality given a graph is similar to writing a linear equation given a graph. Two extra things must be considered.

Whether the line is solid or dashed
Shaded region

Consider the following graph.

The procedure begins by writing the equation of the boundary line in slope-intercept form. To do so, the

y -

intercept and the slope must be found. Then, identify whether the inequality is strict. Finally, the inequality symbol should be determined.

Writing the Linear Equation

To write the linear equation of the boundary line, the slope and $y -$ intercept should be identified. It is easier to do so if the shaded region is ignored.

The

y -

intercept is

2 .

The slope of the line is the quotient of the rise and run.

m = \frac{- 3}{1} \Leftrightarrow m = - 3

Having found the slope and the

y -

intercept, the linear equation can be written in slope-intercept form.

y = - 3 x + 2

Strict or Non-Strict?

To identify whether the inequality is strict or non-strict, the line should be considered. If it is dashed, the inequality is strict. Otherwise, the inequality is non-strict. Consider the boundary line of the given inequality.

The graph has a dashed line. Therefore, the inequality is strict. This means that the inequality sign is either $<$ or $> .$

Inequality Sign

To determine the sign of the inequality, a point located in the shaded region but not on the boundary line should be tested. For simplicity, the point $(1, 1)$ will be used.

The coordinates of this point can be substituted to determine the symbol of the inequality.

y ? - 3 x + 2

SubstituteII

$x = 1$ , $y = 1$

1 ? - 3 (1) + 2

IdPropMult

Identity Property of Multiplication

1 ? - 3 + 2

AddTerms

Add terms

1 > - 1

The sign of the inequality is greater than. With this information, the inequality can be written.

y > - 3 x + 2

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