Graph the system of inequalities and show the solution set. $$\left\{ \begin{array} { l }
y \leq - x - 1 \
y \geq 2 x - 2
\end{array} \right.$$

Question

Graph the system of inequalities and show the solution set. $$\left\{ \begin{array} { l } 
y \leq - x - 1 \
y \geq 2 x - 2
\end{array} \right.$$

Quizplus · Accepted Answer

To graph the system of inequalities, we first graph each individual inequality on the same coordinate plane.

The first inequality, $y \leq -x - 1$, can be graphed by first finding its $x$-intercept at $(-1,0)$ and its $y$-intercept at $(0,-1)$.

[asy]
size(150);
import graph;

real y1 (real x) {return -x - 1;}

Label f;

f.p=fontsize(6);

xaxis(-5,5,Ticks(f, 2.0));

yaxis(-5,5,Ticks(f, 2.0));

draw(graph(y1,-5,4), Arrows(4));
[/asy]

Next, we graph the second inequality, $y \geq 2x - 2$, by first finding its $x$-intercept at $(1,0)$ and its $y$-intercept at $(0,-2)$.

[asy]
size(150);
import graph;

real y2 (real x) {return 2x-2;}

Label f;

f.p=fontsize(6);

xaxis(-5,5,Ticks(f, 2.0));

yaxis(-5,5,Ticks(f, 2.0));

draw(graph(y2,-2,5), Arrows(4));
[/asy]

To find the solution set of the system of inequalities, we shade the region that satisfies both inequalities. The shaded region is the area that is below the line $y=-x-1$ and above the line $y=2x-2$:

[asy]
size(150);
import graph;

real y1 (real x) {return -x - 1;}
real y2 (real x) {return 2x-2;}

Label f;

f.p=fontsize(6);

xaxis(-5,5,Ticks(f, 2.0));

yaxis(-5,5,Ticks(f, 2.0));

filldraw((1,0)--(0,-2)--(-2,-6)--(-5,-4)--(-5,5)--(5,5)--(5,-2)--cycle, gray(.7));

draw(graph(y1,-5,4), Arrows(4));
draw(graph(y2,-2,5), Arrows(4));
[/asy]

The solution set is the shaded region shown above, which corresponds to choice $\boxed{	extbf{(B)}}$.

Graph the System of Inequalities and Show the Solution Set {y≤−x−1y≥2x−2\left\{ \begin{array} { l } y \leq - x - 1 \\y \geq 2 x - 2\end{array} \right.{y≤−x−1y≥2x−2​

Graph the System of Inequalities and Show the Solution Set $\left\{ \begin{array} { l } y \leq - x - 1 \\y \geq 2 x - 2\end{array} \right.$