Q. 400 - Intermediate Algebra

Question

Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.

$\left\{\begin{array}{l}x-y>-1\\ y<\frac{1}{3}x-2\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system of the inequality $\left\{\begin{array}{l}x-y>-1\\ y<\frac{1}{3}x-2\end{array}\right.$ is the overlapped region that contains the point $(0,-3)$

Step 1. Given

The system of inequality is $\left\{\begin{array}{l}x-y>-1\\ y<\frac{1}{3}x-2\end{array}\right.$

To find the solution of inequality by graphing

Step 2. Graph the first inequality

Graph the line $x-y=-1$

It is a dashed line since it contains the inequality $>$.

And test the point $(0,0)$.

It is a solution to the given inequality, so shade the region that contains the point $(0,0)$

Step 3. Graph the second inequality

Graph the line $y=\frac{1}{3}x-2$ 

      with slope $m=\frac{1}{3}$ and $y-$intercept $-2$

The boundary line is dashed line since it contains the inequality $<$.

Test the inequality with point $(0,0)$

It is not a solution so shade the region that does not contain the point $(0,0)$.

Step 4. Solution of the inequality

The point where the boundary line intersect is not a solution since it is not a solution to both the inequalities.

The solution is all the points in the area shaded twice which appears as the darkest shaded region.

Step 5. Check the solution by choosing the point

Choose $(0,-3)$ as a test point

Test for first inequality:

       $x-y>-1$

$0-(-3)>-1$

            $3>-1$ is true.

Test for second inequality:

    $y<\frac{1}{3}x-2$

$-3<\frac{1}{3}\left(0\right)-2$

$-3<-2$ is true.

The region containing $(0,-3)$ is the solution to the system.