Question

In the following exercises, solve the following systems by graphing.

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

Short answer

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

Step-by-step solution

Step 1. Given

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

To find the solution of inequality by graphing

Step 2. Graph the first inequality

Graph the line$x-y=-2$

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 $y\le 3x+1$

It is a solid line since it contains the inequality $\le$.

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 4. Solution of the inequality

The point where the boundary line intersect is not a solution since it is not a solution to the inequality $x-y>-2$.

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 $(0,0)$as a test point

Test for first inequality:

$x-y>-2$

$0>-2$is true.

Test for second inequality:

$y\le 3x+1$

$0\le 3\left(0\right)+1$

$0\le 1$is true.

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