Q 4.109 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution of the system of the inequality $\left\{\begin{array}{l}x+y\le 2\\ y\ge \frac{2}{3}x-1\end{array}\right.$ by graphing is the region that contains the point $(0,0)$

Step 1. Given

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

To solve the inequality by graphing.

Step 2. Graph the first inequality

Graph the equation $x+y=2$ for the inequality $x+y\le 2$

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

The intercepts are $x=2$

                                $y=2$

Test the point $(0,0)$ which makes the inequality true. So shade the region that contains the point

$(0,0)$

Step 3. Graph the second inequality

Graph $y\ge \frac{2}{3}x-1$ by graphing $y=\frac{2}{3}x-1$ using the slope $y=\frac{2}{3}$ and $y-$intercept $1$.

The boundary line is solid.

Test $(0,0)$ which makes the inequality true, so
shade the side that contains $(0,0)$.

Step 4. Test the inequalities with point ( 0 , 0 )

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

By testing the point $(0,0)$ on both the inequalities, they hold true.

So the region that contains $(0,0)$ is the solution to both inequalities