Question

question 3
$y < -x + 4$
$y \ge 0.5x + 3$
o a) graph with coordinate plane, dashed line, solid line, and shaded regions
o b) partial graph with coordinate plane, dashed line, solid line, and shaded region

Explanation:

Step1: Analyze the first inequality \( y < -x + 4 \)

The line \( y=-x + 4 \) has a slope of \(- 1\) and a \(y\)-intercept of \(4\). Since the inequality is \(y < -x + 4\), the line should be dashed (because it's a strict inequality) and we shade below the line.

Step2: Analyze the second inequality \( y\geq0.5x + 3 \)

The line \(y = 0.5x+3\) has a slope of \(0.5\) (or \(\frac{1}{2}\)) and a \(y\)-intercept of \(3\). Since the inequality is \(y\geq0.5x + 3\), the line should be solid (because of the "or equal to" part) and we shade above the line.

Step3: Check the intersection and the shaded regions

• For the line \(y=-x + 4\) (dashed line), when \(x = 0\), \(y = 4\); when \(y=0\), \(x = 4\). The region \(y < -x + 4\) is below this dashed line.
• For the line \(y = 0.5x+3\) (solid line), when \(x = 0\), \(y = 3\); when \(y = 0\), \(x=- 6\). The region \(y\geq0.5x + 3\) is above this solid line.

Now, let's check the options:

• Option A: The dashed line is \(y=-x + 4\) (correct slope and \(y\)-intercept, dashed), and the solid line is \(y = 0.5x+3\) (correct slope and \(y\)-intercept, solid). The shaded region is the intersection of the region below the dashed line and above the solid line, which matches the inequalities.
• Option B: The \(y\)-intercept of the solid line seems incorrect (not \(3\)) and the shaded region does not match the intersection of the two inequalities.

Answer:

A. The graph with the dashed line \(y = -x + 4\) (slope - 1, \(y\)-intercept 4) and solid line \(y=0.5x + 3\) (slope 0.5, \(y\)-intercept 3) with the correct shaded regions (below the dashed line and above the solid line)