solve the following system of inequalities graphically on the set of axes below. state the coordinates of a…

Explanation:

Step1: Find the boundary - line equations

The boundary - line for $y\leq x + 7$ is $y=x + 7$, which has a slope of $m_1 = 1$ and a $y$ - intercept of $b_1=7$. The boundary - line for $y\lt - 2x-8$ is $y=-2x - 8$, which has a slope of $m_2=-2$ and a $y$ - intercept of $b_2=-8$.

Step2: Determine the type of line and shading for $y\leq x + 7$

Since the inequality is $y\leq x + 7$, the boundary line $y=x + 7$ is a solid line. To find which side to shade, we can test a point. Let's test the origin $(0,0)$. Substitute $x = 0$ and $y = 0$ into $y\leq x + 7$: $0\leq0 + 7$ (True). So, we shade the region below the line $y=x + 7$.

Step3: Determine the type of line and shading for $y\lt - 2x-8$

Since the inequality is $y\lt - 2x-8$, the boundary line $y=-2x - 8$ is a dashed line. Test the origin $(0,0)$: Substitute $x = 0$ and $y = 0$ into $y\lt - 2x-8$, we get $0\lt-8$ (False). So, we shade the region below the line $y=-2x - 8$.

Step4: Find the intersection point of the two boundary - lines

Set $x + 7=-2x-8$. Add $2x$ to both sides: $x+2x + 7=-8$. Then $3x+7=-8$. Subtract 7 from both sides: $3x=-8 - 7=-15$. Divide both sides by 3: $x=-5$. Substitute $x = - 5$ into $y=x + 7$, we get $y=-5 + 7 = 2$. The intersection point of the two lines is $(-5,2)$.

Step5: Find a point in the solution set

By observing the shaded region where the two shaded areas overlap, we can choose a point. Let's take $x=-6$. For the first inequality $y\leq x + 7$, $y\leq-6 + 7=1$. For the second inequality $y\lt-2x-8$, when $x=-6$, $y\lt-2\times(-6)-8=12 - 8 = 4$. A point in the solution set could be $(-6,0)$.

Answer:

A point in the solution set is $(-6,0)$