Question

a \

$$\begin{cases} y > 2x - 1 \\\\ y \\leq -2x + 3 \\end{cases}$$

b \

$$\begin{cases} y < 2x - 1 \\\\ y \\geq -2x + 3 \\end{cases}$$

c \

$$\begin{cases} y \\leq 2x - 1 \\\\ y > 2x + 3 \\end{cases}$$

d \

$$\begin{cases} y \\leq 2x - 1 \\\\ y > -2x + 3 \\end{cases}$$

(graph of a system of linear inequalities with two lines, one solid and one dashed, and a shaded region.)

Explanation:

Step1: Analyze the solid line

The solid line has a slope of -2 and a y-intercept of 3 (from the equation \(y = -2x + 3\)). Since it's a solid line, the inequality should include equality, so the inequality is \(y \leq -2x + 3\) (we check the shaded region: the shaded area is below this line, so \(\leq\)).

Step2: Analyze the dashed line

The dashed line has a slope of 2 and a y-intercept of -1 (from the equation \(y = 2x - 1\)). Since it's a dashed line, the inequality does not include equality, so the inequality is \(y > 2x - 1\) (we check the shaded region: the shaded area is above this dashed line, so \(>\)).

Step3: Match with options

Now we check the options. Option A has \(y > 2x - 1\) and \(y \leq -2x + 3\), which matches our analysis.

Answer:

A. \(

$$\begin{cases} y > 2x - 1 \\ y \leq -2x + 3 \end{cases}$$

\)