Question

$x < 2.5$
$-\frac{1}{4}x + \frac{1}{2}y geq 2$

Explanation:

Step1: Analyze \( x < 2.5 \)

The inequality \( x < 2.5 \) represents the region to the left of the vertical dashed line \( x = 2.5 \) (since it's a strict inequality, the line is dashed).

Step2: Analyze \( -\frac{1}{4}x+\frac{1}{2}y\geq2 \)

First, rewrite the inequality in slope - intercept form (\( y = mx + b \)):
\[

$$\begin{align*} -\frac{1}{4}x+\frac{1}{2}y&\geq2\\ \frac{1}{2}y&\geq\frac{1}{4}x + 2\\ y&\geq\frac{1}{2}x+4 \end{align*}$$

\]
The boundary line \( y=\frac{1}{2}x + 4 \) has a slope of \( \frac{1}{2} \) and a y - intercept of 4. Since the inequality is \( \geq \), the line is solid and we shade above the line.

Step3: Analyze the regions in each option

• Option A: The vertical line for \( x < 2.5 \) is at \( x = 3 \) (incorrect, should be \( x = 2.5 \)), and the shading for the linear inequality is not correct.
• Option B: The vertical line for \( x < 2.5 \) is at \( x = 3 \) (incorrect), and the shading for the linear inequality and the combination is wrong.
• Option C: The vertical dashed line is at \( x = 2.5 \) (correct for \( x < 2.5 \)), and the shading above the line \( y=\frac{1}{2}x + 4 \) (with the correct slope and y - intercept) and the combination of the two regions (left of \( x = 2.5 \) and above the linear line) is correct.
• Option D: The shading for the linear inequality and the vertical line region is incorrect.

Answer:

C