Question

what does the shaded region of a linear inequality represent?

• only one solution
• the boundary line only
• the slope of the line
• all possible solutions

Explanation:

Define linear inequalities in a plane

Using the Linear Inequalities knowledge point
An inequality in two variables divides the coordinate plane into two half-planes separated by a boundary line.

Analyze the meaning of the shaded region

Using the Linear Inequalities knowledge point
The shaded region represents the set of all ordered pairs \((x, y)\) that make the inequality statement true. Therefore, it represents all possible solutions to the inequality.

Evaluate the given options

Using the Linear Inequalities knowledge point

• "Only one solution" is incorrect because there are infinitely many points in the shaded region.
• "The boundary line only" is incorrect because the boundary line is just the edge, and the shaded region covers an entire half-plane.
• "The slope of the line" is incorrect because the slope is a single numerical value describing steepness, not a region.
• "All possible solutions" is correct because every point in the shaded region satisfies the inequality.

Answer:

• Only one solution
• The boundary line only
• The slope of the line
• All possible solutions (Correct answer)