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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.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.

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

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.Write the inequality in slope-intercept form

$9x+3y\leq6$

$3y\leq \text{-}9x+6$

$y \leq \dfrac{\text{-} 9x+6}{3}$

$y \leq \text{-} \dfrac{9x}{3} + \dfrac{6}{3}$

$y\leq\text{-} 3x+2$

Graph the boundary line

Test a point

$y\leq \text{-} 3x+2$

${\color{#009600}{0}}\stackrel{?}{\leq} \text{-} 3\cdot{\color{#0000FF}{0}} + 2$

$0\leq 2$

Shade the correct region

If the test point is a solution to the inequality, the region in which it lies contains the entire solution set. If not, the other region contains the solutions. To show the set, shade the appropriate region.

Here, test point $(0,0)$ is a solution to the inequality.

The region containing $(0,0)$ lies to the left of the boundary line. Thus, this region shows the solution set of the inequality.

Is the point $(4,\text{-}2)$ a solution to the inequality $2y-x>6?$

A point is a solution to a linear inequality if, when substituting the values of $x$ and $y,$ a true statement is made. Here, we can substitute $x=4$ and $y=\text{-} 2$ into the given inequality and simplify.
$\text{-}8$ is not greater than $6,$ so $(4,\text{-}2)$ is **not** a solution to the inequality. We can verify our answer by graphing the inequality and the point.

$2y-x>6$

$2({\color{#009600}{\text{-}2}})-{\color{#0000FF}{4}} \stackrel{?}{>} 6$

$\text{-}4-4\stackrel{?}{>}6$

$\text{-}8\ngtr6$

The point does not lie inside the shaded region, so it's not part of the solution set.

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