Question

39 which graph best represents the solution set of inequalities?
$x + 2y < -2$
$y - x < 3$
(there are four graphs labeled a, b, c, d with coordinate planes and lines/shaded regions)

Explanation:

Step1: Analyze first inequality

Rewrite \(x + 2y < -2\) as \(y < -\frac{1}{2}x - 1\). The boundary line is \(y = -\frac{1}{2}x - 1\) (slope \(-\frac{1}{2}\), y - intercept \(-1\)), dashed (since \(<\)), and shade below.

Step2: Analyze second inequality

Rewrite \(y - x < 3\) as \(y < x + 3\). The boundary line is \(y = x + 3\) (slope \(1\), y - intercept \(3\)), dashed (since \(<\)), and shade below.

Step3: Match with graphs

Check each graph:

• Graph A: Solid lines (incorrect, inequalities are strict, so dashed).
• Graph B: Lines don't match slopes/intercepts.
• Graph C: Dashed lines, \(y = -\frac{1}{2}x - 1\) (slope \(-\frac{1}{2}\), y - intercept \(-1\)) and \(y = x + 3\) (slope \(1\), y - intercept \(3\)), shading below both. Matches.
• Graph D: Solid lines (incorrect).

Answer:

C