Question

question 25 · 1 point
graph the following system of linear inequalities to find the solution set. drag the points into
the correct positions to determine the lines and then drag the shading points to select the
correct shading.

y ≥ (x/2) - 2
y < 4x + 4

provide your answer below:

graph with shading options and grid, lines, points

reset

Explanation:

Step1: Analyze \( y \geq \frac{x}{2} - 2 \)

The line \( y = \frac{x}{2} - 2 \) has a slope of \( \frac{1}{2} \) and a y - intercept of \( - 2 \). Since the inequality is \( \geq \), the line should be solid, and we shade above the line.

Step2: Analyze \( y < 4x + 4 \)

The line \( y = 4x + 4 \) has a slope of \( 4 \) and a y - intercept of \( 4 \). Since the inequality is \( < \), the line should be dashed, and we shade below the line.

Step3: Identify the lines and shading

• For \( y=\frac{x}{2}-2 \): Let's find two points on the line. When \( x = 0 \), \( y=-2 \); when \( x = 4 \), \( y=\frac{4}{2}-2=0 \). The blue line (with a smaller slope) should represent \( y = \frac{x}{2}-2 \) (solid line as per the inequality \( \geq \)).
• For \( y = 4x + 4 \): When \( x = 0 \), \( y = 4 \); when \( x=-1 \), \( y=4\times(-1)+4 = 0 \). The red line (with a larger slope) should represent \( y = 4x + 4 \) (dashed line as per the inequality \( < \)).
• The shading region should be the area that is above the solid blue line (\( y\geq\frac{x}{2}-2 \)) and below the dashed red line (\( y < 4x + 4 \)). Looking at the shading options, the correct shading should be the region that satisfies both inequalities. The intersection of the two shaded regions (above the blue line and below the red line) is the solution set. The correct shading points and line positions should be adjusted such that the blue line is \( y=\frac{x}{2}-2 \) (solid, passing through (0, - 2) and (4,0)) and the red line is \( y = 4x+4 \) (dashed, passing through (0,4) and (- 1,0)), and the shading is in the region that is above the blue line and below the red line.

Answer:

To graph the system:

1. Draw the solid line \( y=\frac{x}{2}-2 \) (slope \( \frac{1}{2} \), y - intercept - 2) and shade above it.
2. Draw the dashed line \( y = 4x + 4 \) (slope 4, y - intercept 4) and shade below it.
3. The solution set is the region that is in both shaded areas (above \( y=\frac{x}{2}-2 \) and below \( y = 4x + 4 \)). The correct graphing involves positioning the blue (solid) line as \( y=\frac{x}{2}-2 \), the red (dashed) line as \( y = 4x + 4 \), and shading the overlapping region (above the blue solid line and below the red dashed line).