find the graph of this system of linear inequalities. \\(\\begin{cases} y \\leq x + 1 \\\\ y < -\\frac{x}{2}…

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})