Question

which graph shows the solution to this system of inequalities?
$4x - 2y \geq -4$
$2x + y > -3$
answer options
graph w graph x
graph y graph z
a. graph w
b. graph x
c. graph y
d. graph z

Explanation:

Step1: Rewrite inequalities in slope - intercept form

For the first inequality \(4x - 2y\geq - 4\), we solve for \(y\):
Subtract \(4x\) from both sides: \(-2y\geq - 4x - 4\)
Divide both sides by \(- 2\) (and reverse the inequality sign): \(y\leq2x + 2\)
This is a line with slope \(m = 2\) and \(y\) - intercept \(b = 2\). The inequality \(y\leq2x + 2\) means we shade below the line (and the line is solid because of the \(\geq\) sign in the original inequality, but after manipulation, it's \(\leq\), so the line for \(y = 2x+2\) should be solid? Wait, no, let's re - check:
Starting with \(4x-2y\geq - 4\)
\(-2y\geq - 4x - 4\)
Divide by \(-2\): \(y\leq2x + 2\). So the boundary line \(y = 2x + 2\) is solid, and we shade below it.

For the second inequality \(2x + y>-3\), solve for \(y\): \(y>-2x - 3\)
This is a line with slope \(m=-2\) and \(y\) - intercept \(b = - 3\). The inequality \(y>-2x - 3\) means we shade above the line, and the line is dashed (because of the \(>\) sign).

Step2: Analyze the boundary lines and shading

• The line \(y = 2x+2\) (from \(4x - 2y\geq - 4\)) has a positive slope and \(y\) - intercept at \(y = 2\). The line \(y=-2x - 3\) (from \(2x + y>-3\)) has a negative slope and \(y\) - intercept at \(y=-3\).
• For the first inequality (\(y\leq2x + 2\)), we shade below the solid line. For the second inequality (\(y>-2x - 3\)), we shade above the dashed line.
• Now let's check the graphs:
• Graph W: The solid line has a positive slope (matches \(y = 2x + 2\)), the dashed line has a negative slope (matches \(y=-2x - 3\)). The shading is above the dashed line and below the solid line, which is the intersection of the two solution regions.
• Graph X: The shading seems to be in a triangular region that does not match the correct intersection of the two solution sets.
• Graph Y: The shading is in a region that is below both lines, which does not match the solution of \(y>-2x - 3\) (which requires shading above the dashed line).
• Graph Z: The shading is above the solid line, which does not match \(y\leq2x + 2\) (which requires shading below the solid line).

Answer:

A. Graph W