Q 4.110 - Intermediate Algebra

Question

Solve the system by graphing. $\left\{\begin{array}{l}3x-2y\le 6\\ y>-\frac{1}{4}x+5\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system of the inequality $\left\{\begin{array}{l}3x-2y\le 6\\ y>-\frac{1}{4}x+5\end{array}\right.$ by graphing is the region that contains the point $(4,5)$

Step 1. Given

The system of inequality is $\left\{\begin{array}{l}3x-2y\le 6\\ y>-\frac{1}{4}x+5\end{array}\right.$

To solve the inequality by graphing.

Step 2. Graph the first inequality

Graph the equation $3x-2y=6$ for the inequality $3x-2y\le 6$

And test the point $(0,0)$ and it holds true.

The intercepts are $x=2$,

                                $y=-3$

Since the test point $(0,0)$ makes the inequality true, shade the region that contains the point.

Step 3. Graph the second inequality

Graph $y>-\frac{1}{4}x+5$ by graphing $y=-\frac{1}{4}x+5$ using the slope $m=-\frac{1}{4}$ and the $y-$intercept $5$.

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 $\left\{\begin{array}{l}3x-2y\le 6\\ y>-\frac{1}{4}x+5\end{array}\right.$

By testing the point $(4,5)$ on both inequalities,

they hold true.

So the region that contains $(4,5)$ is the solution to both inequalities