Q 4.108 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

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.