Question

the shaded region shown represents the solutions to the inequality -16y < c, where c is a constant. what is the value of c? (a) -96 (b) -6 (c) 6 (d) 96

Explanation:

Step1: Identify the boundary line

The shaded region has a horizontal boundary line. From the graph, the boundary line is \( y = -6 \) (since the horizontal line is at \( y = -6 \)). For the inequality \( -16y < c \), the boundary occurs when \( -16y = c \).

Step2: Substitute y = -6 into the equation

Substitute \( y = -6 \) into \( -16y = c \). So, \( c = -16\times(-6) \).

Step3: Calculate c

Calculate \( -16\times(-6) = 96 \)? Wait, no, wait. Wait, the inequality is \( -16y < c \). Let's solve for y: \( y > -\frac{c}{16} \) (since dividing by a negative number reverses the inequality). The boundary line is \( y = -6 \), so \( -\frac{c}{16} = -6 \). Solve for c: Multiply both sides by -16: \( c = (-6)\times(-16) = 96 \)? Wait, no, wait. Wait, let's check again. If \( y > -\frac{c}{16} \), and the boundary is \( y = -6 \), then \( -\frac{c}{16} = -6 \). Multiply both sides by -16: \( c = (-6)\times(-16) = 96 \)? Wait, but the options have D as 96? Wait, no, wait the graph: the shaded region is above y = -6? Wait, the shaded region is above the line y = -6? Wait, the graph shows the shaded area is above y = -6? Wait, the grid: the horizontal line is at y = -6. Wait, the inequality is -16y < c. Let's solve for y: divide both sides by -16 (reverse inequality): y > c/(-16) => y > -c/16. So the boundary is y = -c/16. From the graph, the boundary is y = -6. So -c/16 = -6 => c/16 = 6 => c = 96. Wait, but let's check the options. Option D is 96. Wait, but wait, maybe I made a mistake. Wait, the shaded region: looking at the graph, the shaded area is above y = -6? Wait, the grid: the y-axis, the line is at y = -6. So the boundary is y = -6, and the inequality is y > -6 (since shaded above). So from -16y < c, solving for y: y > c/(-16) (because dividing by -16 reverses inequality). So y > -c/16. So -c/16 = -6 => c/16 = 6 => c = 96. So c is 96. Wait, but let's check the calculation again. -16y < c. Let's take a point in the shaded region, say (0, -5) (which is above y = -6). Plug into -16y < c: -16(-5) = 80 < c? Wait, no, if c is 96, then 80 < 96, which is true. If we take y = -6, then -16(-6) = 96, so 96 < c? No, the boundary is when -16y = c. So the line is -16y = c, which is y = -c/16. So the line is y = -6, so -c/16 = -6 => c = 96. So the value of c is 96.

Answer:

D. 96