Question

solve the system of two linear inequalities graphically.\
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$$\begin{cases} y \\leq -5x + 10 \\\\ y > x - 2 \\end{cases}$$

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step 3 of 3 : graph the solution set for the system.\
answer

Explanation:

Step1: Graph boundary $y=-5x+10$

Draw a solid line (since $\leq$ includes equality) with slope $-5$ and y-intercept $10$.

Step2: Shade region for $y\leq-5x+10$

Shade all areas below the solid line.

Step3: Graph boundary $y=x-2$

Draw a dashed line (since $>$ excludes equality) with slope $1$ and y-intercept $-2$.

Step4: Shade region for $y>x-2$

Shade all areas above the dashed line.

Step5: Identify overlapping region

The solution set is the intersection of the two shaded regions.

Answer:

The solution is the overlapping shaded area: below the solid line $y=-5x+10$ and above the dashed line $y=x-2$. The intersection point of the two boundaries is found by solving $-5x+10 = x-2$:
$$-5x - x = -2 -10$$
$$-6x = -12$$
$$x=2$$
Substitute $x=2$ into $y=x-2$: $y=0$. So the overlapping region is bounded by the two lines, meeting at $(2, 0)$, with the solid line forming the upper boundary and the dashed line forming the lower boundary of the solution area.