Question

which point(s) are solutions to the system graphed below?

(2, -1)
(-6, 2)
(-5, 0)
(3, 4)

Explanation:

Identify the solution region

Using the Systems of Linear Inequalities knowledge point
The solution set of a system of linear inequalities is represented graphically by the overlapping shaded region. In the provided graph, the solution region is the yellow-shaded area in the lower-left quadrant, bounded by:

• A solid red line: \(y \le -\frac{1}{2}x - 1\) (shaded below the solid line)
• A dashed blue line: \(y > 2x - 3\) (shaded to the left/above the dashed line)

Test the given coordinates

Using the Systems of Linear Inequalities knowledge point
We locate each option on the coordinate plane to see if it lies within the yellow shaded region:

• Point \((2, -1)\): This point lies exactly on the intersection of the solid red line and the dashed blue line. Since the blue line is dashed, points on it are not included in the solution set.
• Point \((-6, 2)\): This point lies in the region above the solid red line, which is unshaded.
• Point \((-5, 0)\): This point lies in the yellow shaded region, below the solid red line and to the left of the dashed blue line.
• Point \((3, 4)\): This point lies in the unshaded region in the upper-right quadrant.

Confirm the correct point

Using the Systems of Linear Inequalities knowledge point
Only the point \((-5, 0)\) lies entirely within the shaded solution region.

Answer:

• (A) \((2, -1)\)
• (B) \((-6, 2)\)
• (C) \((-5, 0)\) (Correct answer)
• (D) \((3, 4)\)