Question

Solve the system by graphing. $\left\{\begin{array}{l}y<-\frac{1}{2}x+3\\ y<3x-4\end{array}\right.$

Short answer

The solution of the system $\left\{\begin{array}{l}y<-\frac{1}{2}x+3\\ y<3x-4\end{array}\right.$i

Step-by-step solution

Step 1. Given

The system of inequality equations: $\left\{\begin{array}{l}y<-\frac{1}{2}x+3\\ y<3x-4\end{array}\right.$

To solve the system by graphing.

Step 2. Graph the first inequality

Graph the line $y=-\frac{1}{2}x+3$

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 the region which contains the test point.

Step 3. Graph the second inequality.

Graph the boundary line $y<3x-4$

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

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

It is a solution so we shade the region which contins the test point.

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 $(2,1)$as a test point.

Substitute $(2,1)$in the inequality

$y<-\frac{1}{2}x+3$

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

$1<-1+3$

$1<2$is true

Substitute $(2,1)$in the inequality $y<3x-4$

$1<3\left(2\right)-4$

$1<6-4$

$1<2$is true.

The region containing $(2,1)$is the solution to the system.