Question

Solve the system by graphing. $\left\{\begin{array}{l}y<3x+2\\ y>-x-1\end{array}\right.$

Short answer

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

Step-by-step solution

Step 1. Given

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

To solve the system by graphing.

Step 2. Graph the first inequality y<3x+2

Graph the line $y=3x+2$

It is a dashed line because the inequality sign is $<$.

Shade in the side of that boundary line where the inequality is true.

Use $(0,0)$as a test point.

It is a solution so we shade in that side of the line $y=3x+2$

Step 3. Graph the second inequality.

Graph the boundary line $y=-x-1$.

It is a dashed line because the inequality sign is $>$

Use $(0,0)$as a test point.

It is a solution so we shade in that side of the line $y=-x-1$

Step 4. Find the region of intersection

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

Step 5. Check by using test point

Choose $(1,1)$as a test point.

Substitute $(1,1)$in the inequality $y<3x+2$

$1<3\left(1\right)+2$

$1<5$is true.

Substitute $(1,1)$in the inequality $y>-x-1$

$1>-1-1$

$1>-2$is true

The region containing $(1,1)$ is the solution to the system.