Question

Graph each inequality.

14.$3\ge x-3y$.

Short answer

The graph of the inequality is shown below:

Step-by-step solution

– Find the boundary line equation of the given equation.

Consider the inequality$3\ge x-3y$.

The boundary line equaton is $x-3y=3$.

Here the inequality is$\ge$, graph the boundary line as a solid line.

Find the slope intercept of$x-3y=3$:

$\begin{array}{l}x-3y=3\\ -3y=3-x\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}subtract}x\text{frombothsides}\\ y=\frac{3-x}{-3}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}dividebothsidesby}-3\\ y=\frac{x}{3}-1\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}simplify}\end{array}$

So, the slope intercept form is $y=\frac{x}{3}-1.$

– Choose a test point and test the inequality.

Choose the test point as $\left(0,0\right).$

Substitute $x=0,y=0in3\backslash gex-3y$

$\begin{array}{l}3\ge x-3y\\ 3\ge 0-3\left(0\right)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}substitute0for}x,0\text{for}y\\ 3\ge 0\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}True}\end{array}$

– Conclusion

Since, $3\ge 0$results a true inequality, so, shade the region that contains the point $\left(0,0\right).$

Graphical representation of$3\backslash gex-3y$