Question

If the system of inequalities $y\ge 2x+1$ and $y>\frac{1}{2}x-1$ is graphed in the $xy$-plane above, which quadrant contains no solutions to the system?

• A. Quadrant II
• B. Quadrant III
• C. Quadrant IV
• D. There are solutions in all four quadrants

Answer 1

The correct option is C Quadrant IV

given, $y\ge 2x+1$ and

$y>\frac{1}{2}x-1$

first, draw the graph for equations $y=2x+1$ and $y=\frac{1}{2}x-1$

for $y=2x+1$

substitute y=0 we get, $2x+1=0⟹x=-0.5$

substitute x=0 we get, $y=1$

therefore,$y=2x+1$ line passes through (0.5,0) and (0,1) as shown in fig.

Hence, $y\ge 2x+1$ includes the region above the line.

for $y=\frac{1}{2}x-1$

substitute y=0 we get, $\frac{1}{2}x-1=0⟹x=2$

substitute x=0 we get, $y=-1$

therefore,$y=2x+1$ line passes through (2,0) and (0,-1) as shown in fig.

Hence, $y>\frac{1}{2}x-1$includes the region above the line.

the intersection region is the shaded region as shown in above figure which includes I, II and III quadrants.

Therefore, quadrant IV has no solution