Question

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

$\left\{\begin{array}{l}2x-3y<6\\ 3x+4y\ge 12\end{array}\right.$

Short answer

The solution of the system of inequality $\left\{\begin{array}{l}2x-3y<6\\ 3x+4y\ge 12\end{array}\right.$is the overlapped region that contains the point

Step-by-step solution

Step 1. Given

The system of inequality is$\left\{\begin{array}{l}2x-3y<6\\ 3x+4y\ge 12\end{array}\right.$

To find the solution of inequality by graphing

Step 2. Graph the first inequality

Graph the line$2x-3y=6$

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 3. Graph the second inequality

Graph the line$3x+4y=12$

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

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

It is not a solution to the given inequality, so shade the region that does not contain 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 $2x-3y<6$.

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 $(0,4)$as a test point

Test for first inequality:

$2x-3y<6$

$2\left(0\right)-3\left(4\right)<6$

$-12<6$is true.

Test for second inequality:

$3x+4y\ge 12$

$3\left(0\right)+4\left(4\right)\ge 16$

$16\ge 16$is true.

The region containing $(0,4)$is the solution to the system.