Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.$y \geq -x + 8$$y \leq \frac{3}{2}x - 7$answer attempt 1 out of 2point: (□, □)submit answer

Explanation:

Step1: Find intersection point

Set $-x + 8 = \frac{3}{2}x - 7$

Step2: Solve for x

$-x - \frac{3}{2}x = -7 - 8$
$-\frac{5}{2}x = -15$
$x = (-15) \times (-\frac{2}{5}) = 6$

Step3: Solve for y

Substitute $x=6$ into $y=-x+8$:
$y = -6 + 8 = 2$

Step4: Verify a solution point

Test $(0,0)$:
For $y \geq -x+8$: $0 \geq 8$ is false.
For $y \leq \frac{3}{2}x-7$: $0 \leq -7$ is false.
Test $(6,2)$:
$2 \geq -6+8=2$ (true), $2 \leq \frac{3}{2}(6)-7=9-7=2$ (true).
Test $(8,0)$:
$0 \geq -8+8=0$ (true), $0 \leq \frac{3}{2}(8)-7=12-7=5$ (true).

Answer:

A point in the solution set is $\boldsymbol{(6, 2)}$ (or other valid points like $(8,0)$)

Graphing Notes (for the axes):

1. For $y \geq -x+8$: Plot the line $y=-x+8$ (y-intercept $(0,8)$, x-intercept $(8,0)$), shade above the line.
2. For $y \leq \frac{3}{2}x-7$: Plot the line $y=\frac{3}{2}x-7$ (y-intercept $(0,-7)$, x-intercept $(\frac{14}{3},0)$), shade below the line.
3. The overlapping shaded region is the solution set.