# Graphing Linear Inequalities

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## Linear Inequality

A linear inequality is an inequality involving a linear relation in one or two variables, usually $x$ and $y.$ An example of a linear inequality is $9x+3y\leq6.$

Linear inequalities are similar to linear equations, but, whereas the solutions to a linear equation are all the coordinates that lie on the line, the solution set to a linear inequality is a region containing one half of the coordinate plane.## Why does the graph of an inequality contain a region?

The solutions of a linear equation form a line in a coordinate plane. Linear **inequalities,** on the other hand, are sets of coordinates that create an entire region of a coordinate plane. This begs the question $``$Why does the graph of an inequality contain a region?" Consider the following inequality.
$y \geq x$
The boundary line to the inequality is $y = x.$ It's the line that passes through all points where $x$ and $y$ have the same value. These include $(\text{-}1,\text{-}1),$ $(0,0),$ $(1,1),$ etc.

**less than or equal to**the $x$-value. For $x=4,$ the inequality becomes $y \leq 4.$ Thus, for all points with $x=4,$ if the corresponding $y$-value is less than or equal to $4,$ the point is a solution to the inequality.

The reasoning can be applied to several $x$-values. Applying it to **all** $x$-values creates the entire region below the line $y=x.$

## Graphing a Linear Inequality

The method to graph a linear inequality is similar to graphing a linear equation in slope-intercept form, but instead of a line, the graph of a linear inequality is an entire region.

To graph the linear inequality $9x+3y\leq6,$ write the inequality in slope-intercept form, draw the boundary line, and shade the region that contains the solutions.## Exercises

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