Question

Graph each inequality.

21.$4x-5y-10\le 0$.

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 $4x-5y-10\le 0$.

The boundary line equaton is $4x-5y-10=0$.

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

Find the slope intercept form of $4x-5y-10=0$:

$\begin{array}{l}4x-5y-10=0\\ 4x-5y=10\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Add10onbothsides}\\ -5y=-4x+10\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Subtract}4x\text{frombothsides}\\ y=\frac{-4x+10}{-5}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Dividebothsidesby}-5\\ y=\frac{4}{5}x-2\end{array}$

The slope intercept form is $y=\frac{4}{5}x-2$with slope $\frac{4}{5}.$

– Choose a test point and test the inequality.

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

Substitute $x=0,y=0$in $4x-5y-10\le 0$:

$\begin{array}{l}4x-5y-10\le 0\\ 4\left(0\right)-5\left(0\right)-10\le 0\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}substitute0for}x,0\text{for}y\\ -10\le 0\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}true}\end{array}$

– Conclusion

Since, $-10\le 0$results a true inequality, so, shade the region that contains the point $\left(0,0\right).$

Graphical representation of $4x-5y-10\le 0$: