Question

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

Short answer

The solution is the area shaded twice—which appears as the darkest shaded region which contains the point $(0,-2)$

Step-by-step solution

Step 1. Given

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

To find the solution by graphing

Step 2. Graph the first inequality

Graph $y\ge 3x-2$by graphing $y=3x-2$and the intercepts are $x=\frac{2}{3}$,

$y=-1$

Test a point $(0,0)$.

And it holds true. So shade the region that contains the inequality.

Step 3. Graph the second inequality

Graph $y<-1$by graphing $y=-1$.

It is a horizontal line through $y=-1$and the boundary line will be dashed.

Test $(0,0)$that makes the inequality false.

So shade the region that does not contain the test point.

Step 4.Test the inequalities with point in the overlapped region 

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

Test the point $(0,-2)$and it holds true for both the region.

So the overlapped region that contains $(0,-2)$ is the solution to the system.