Question

consider this system of inequalities.
$3x + y \geq 7$
$x - 2y < -4$
select the graph of the solution to the system.

Explanation:

Step1: Analyze \(3x + y \geq 7\)

Rewrite it as \(y \geq - 3x + 7\). The boundary line is \(y=-3x + 7\) (solid line, since \(\geq\)), and we shade above the line.

Step2: Analyze \(x - 2y < - 4\)

Rewrite as \(-2y < -x - 4\), then \(y>\frac{1}{2}x + 2\) (dashed line, since \(<\)), and we shade above the line (because when we divide by negative, inequality sign flips).

Step3: Check the intersection

Find a test point, e.g., \((0,0)\):

• For \(3x + y \geq 7\): \(0\geq7\) is false, so \((0,0)\) is not in this region.
• For \(x - 2y < - 4\): \(0< - 4\) is false, so \((0,0)\) is not in this region.

Now, check the options:

• Option A: Shaded region includes \((0,0)\) (false for both inequalities), so no.
• Option B: Shaded region for \(3x + y \geq 7\) is above the line (correct), and for \(x - 2y < - 4\) is above the dashed line (correct). Test a point in B's shaded region, e.g., \((3,5)\):
• \(3(3)+5 = 14\geq7\) (true).
• \(3 - 2(5)=3 - 10=-7 < - 4\) (true).
• Option C: Shaded region for \(3x + y \geq 7\) includes \((0,0)\) (false), so no.
• Option D: Shaded region for \(x - 2y < - 4\) is below the dashed line (false), so no.

Answer:

B. The graph with the solid line \(3x + y = 7\) (shaded above) and dashed line \(x - 2y=-4\) (shaded above their intersection).