Question

find the graph of this system of linear inequalities. \\(\

$$\begin{cases} y \\leq x + 1 \\\\ y < -\\frac{x}{2} - 1 \\end{cases}$$

\\)

Explanation:

Step1: Analyze \( y \leq x + 1 \)

The inequality \( y \leq x + 1 \) represents the region below (including the line) the line \( y = x + 1 \). The line \( y = x + 1 \) has a slope of \( 1 \) and a y - intercept of \( 1 \). Since the inequality is "less than or equal to", the line should be solid.

Step2: Analyze \( y < -\frac{x}{2}-1 \)

The inequality \( y < -\frac{x}{2}-1 \) represents the region below (not including the line) the line \( y = -\frac{x}{2}-1 \). The line \( y = -\frac{x}{2}-1 \) has a slope of \( -\frac{1}{2} \) and a y - intercept of \( - 1 \). Since the inequality is "less than", the line should be dashed.

Step3: Find the intersection region

We need to find the region that satisfies both inequalities. For \( y \leq x + 1 \), we shade below the solid line \( y=x + 1 \). For \( y<-\frac{x}{2}-1 \), we shade below the dashed line \( y =-\frac{x}{2}-1 \). The correct graph will be the one where the overlapping region (the region that is below both the solid line \( y=x + 1 \) and the dashed line \( y=-\frac{x}{2}-1 \)) is represented correctly. Looking at the three graphs, the first graph (left - most) has the correct shading: the region below \( y = x+1 \) (solid line) and below \( y=-\frac{x}{2}-1 \) (dashed line) with the appropriate line styles (solid for \( y\leq x + 1 \), dashed for \( y<-\frac{x}{2}-1 \)) and the overlapping region.

Answer:

The left - most graph (the first one among the three given graphs) is the graph of the system of linear inequalities \(

$$\begin{cases}y\leq x + 1\\y<-\frac{x}{2}-1\end{cases}$$

\)