Question

which graph shows the solution to the system of linear inequalities?
y ≥ 2x + 1
y ≤ 2x − 2

Explanation:

Step1: Analyze the slopes and intercepts

The two inequalities are \( y \geq 2x + 1 \) and \( y \leq 2x - 2 \). Both lines have a slope of \( 2 \) (since the coefficient of \( x \) is \( 2 \) in both equations). The first line \( y = 2x + 1 \) has a y-intercept of \( 1 \), and the second line \( y = 2x - 2 \) has a y-intercept of \( -2 \).

Step2: Determine the direction of shading

For \( y \geq 2x + 1 \), we shade above the line \( y = 2x + 1 \) (since \( y \) is greater than or equal to the line). For \( y \leq 2x - 2 \), we shade below the line \( y = 2x - 2 \) (since \( y \) is less than or equal to the line).

Step3: Check for overlapping region

Now, we look at the two lines \( y = 2x + 1 \) and \( y = 2x - 2 \). Since they have the same slope (\( 2 \)), they are parallel lines. The line \( y = 2x + 1 \) is above the line \( y = 2x - 2 \) (because \( 1 > -2 \)). So, the region above \( y = 2x + 1 \) and the region below \( y = 2x - 2 \) will never overlap (because a number can't be both greater than or equal to \( 2x + 1 \) and less than or equal to \( 2x - 2 \) at the same time, since \( 2x + 1>2x - 2 \) for all real \( x \)). Therefore, there is no solution to this system of inequalities, and the graph that shows no overlapping shaded region (where the two shaded regions don't intersect) is the correct one. From the given options, the graph with the red and blue regions not overlapping (like the one with the two parallel lines and no common shaded area) is the answer.

Answer:

The graph with no overlapping shaded region (since the two lines are parallel and the inequalities define regions that do not intersect)