Question

if the shaded region in the graph represents the solution set to a system of inequalities, which of the following could be the system?
choose 1 answer:
a $y \geq \frac{3}{2}x - 1$
$y \geq \frac{5}{2}x - 2$
b $y \geq \frac{3}{2}x - 1$
$y \leq \frac{5}{2}x - 2$
c $y \leq \frac{3}{2}x - 1$
$y \geq \frac{5}{2}x - 2$
d $y \leq \frac{3}{2}x - 1$
$y \leq \frac{5}{2}x - 2$

Explanation:

Step1: Identify line equations

First, find the equations of the two lines using two points each.
For the steeper line: passes through (2,3) and (0,-2). Slope $m_1=\frac{3-(-2)}{2-0}=\frac{5}{2}$, so equation is $y=\frac{5}{2}x-2$.
For the less steep line: passes through (2,2) and (0,-1). Slope $m_2=\frac{2-(-1)}{2-0}=\frac{3}{2}$, so equation is $y=\frac{3}{2}x-1$.

Step2: Test inequality direction

Use a test point in the shaded region, e.g., (0,0):

• For $y=\frac{3}{2}x-1$: $0 \geq \frac{3}{2}(0)-1$ → $0\geq-1$, which is true. So inequality is $y\geq\frac{3}{2}x-1$.
• For $y=\frac{5}{2}x-2$: $0 \leq \frac{5}{2}(0)-2$ → $0\leq-2$, which is false. Reverse the sign: $0\geq\frac{5}{2}(0)-2$ is false, wait correction: test point (0,0) is above the steeper line? No, (0,0) is above $y=\frac{5}{2}x-2$ (since -2 < 0), and the shaded region is above the less steep line and below the steeper line? No, looking at the graph: shaded region is between the two lines, above the lower (less steep) line and below the upper (steeper) line. Wait no, (0,0) is in shaded area: for steeper line $y=\frac{5}{2}x-2$, at x=0, y=-2, 0 > -2, but shaded region includes (0,0), and the steeper line's shaded side is below it? Wait no, take x=2: steeper line is y=3, less steep is y=2. Shaded region at x=2 is y between 2 and 3, so $y\geq\frac{3}{2}x-1$ and $y\leq\frac{5}{2}x-2$.

Answer:

B. $y \geq \frac{3}{2}x - 1$
$y \leq \frac{5}{2}x - 2$