Question

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

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

Short answer

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

Step-by-step solution

Step 1. Given

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

To find the solution of inequality by graphing.

Step 2. Graph the first inequality

Graph the line $y=3x+1$.

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

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

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$y=-x-2$

The boundary line is a solid line since it contains the inequality $\ge$.

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

It is a solution 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 $y<3x+1$

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.

$y<3x+1$

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

$0<1$is true.

For the inequality, $y\ge -x-2$,

$0\ge -\left(0\right)-2$

$0\ge -2$is true

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