Q. 404 - Intermediate Algebra

Question

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

$\left\{\begin{array}{l}y\ge 2x-5\\ -6x+3y>-4\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system of the inequality $\left\{\begin{array}{l}y\ge 2x-5\\ -6x+3y>-4\end{array}\right.$ is the overlapped region that contains the point $(0,0)$

Step 1. Given

The system of inequality is $\left\{\begin{array}{l}y\ge 2x-5\\ -6x+3y>-4\end{array}\right.$

To find the solution of inequality by graphing 

Step 2. Graph the first inequality

Graph the line $y=2x-5$

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

And test the point $(0,0)$

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 $-6x+3y=-4$

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

And test the point $(0,0)$.

It is a solution to the given inequality, 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  the inequality $-6x+3y>-4$

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

Test for first inequality:

$y\ge 2x-5$

$0\ge 2\left(0\right)-5$

$0\ge -5$ is true.

Test for second inequality:

     $-6x+2y>-4$

$-6\left(0\right)+2\left(0\right)>-4$

                  $0>-4$ is true.

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