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# Graphing Linear Inequalities

Concept

## 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.
Explanation

## 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.

The inequality $y \leq x$ describes all the points where the $y$-coordinate is 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 $y=x.$

This means the area on and below the line $y=x$ contains all pairs of $x$ and $y$ that satisfy the inequality $y \leq x.$ Thus, an entire region is created.
Method

## 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.

### 1

Write the inequality in slope-intercept form
To find the boundary line of the region, start by writing the inequality in slope-intercept form. In other words, solve for $y.$
$9x+3y\leq6$
$3y\leq \text{-}9x+6$
$y \leq \dfrac{\text{-} 9x+6}{3}$
$y \leq \text{-} \dfrac{9x}{3} + \dfrac{6}{3}$
$y\leq\text{-} 3x+2$
Written in slope-intercept form, the inequality becomes $y\leq \text{-} 3x+2.$

### 2

Graph the boundary line

The boundary line of the inequality is the line corresponding to the equation produced if the inequality symbol is replaced by an equals sign. In this case, this is the line $y=\text{-} 3x+2.$ If the inequality symbol is $<$ or $>$, the boundary line is dashed. If the symbol is $\leq$ or $\geq$, the line is solid. Here, the line will be solid. The boundary line can be graphed using the $y$-intercept and the slope.

### 3

Test a point
The region either to the left or the right of the boundary line contains the solution set. To determine which, substitute an arbitrary test point (not on the boundary line) into the inequality to determine if it is a solution. Using $(0,0)$ is preferable.
$y\leq \text{-} 3x+2$
${\color{#009600}{0}}\stackrel{?}{\leq} \text{-} 3\cdot{\color{#0000FF}{0}} + 2$
$0\leq 2$
Since $0 \leq 2$ makes a true statement, it is a solution to the inequality.

### 4

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.

Exercise

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

Solution
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.
$2y-x>6$
$2({\color{#009600}{\text{-}2}})-{\color{#0000FF}{4}} \stackrel{?}{>} 6$
$\text{-}4-4\stackrel{?}{>}6$
$\text{-}8\ngtr6$
$\text{-}8$ 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.

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

info Show solution Show solution
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