Question

which graph shows the solution to this system of inequalities?
x - 3y ≥ 3
2x - y ≤ 4
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 \(x - 3y\geq3\):
Subtract \(x\) from both sides: \(- 3y\geq - x + 3\)
Divide both sides by \(-3\) (and reverse the inequality sign): \(y\leq\frac{1}{3}x - 1\)
The boundary line is \(y=\frac{1}{3}x - 1\) (solid line, since the inequality is \(\leq\)) and we shade below the line.

For the second inequality \(2x - y\leq4\):
Subtract \(2x\) from both sides: \(-y\leq - 2x + 4\)
Multiply both sides by \(- 1\) (and reverse the inequality sign): \(y\geq2x - 4\)
The boundary line is \(y = 2x-4\) (solid line, since the inequality is \(\geq\)) and we shade above the line.

Step2: Analyze the slope and intercept of boundary lines

• The line \(y=\frac{1}{3}x - 1\) has a slope of \(\frac{1}{3}\) (positive, gentle slope) and a \(y\) - intercept of \(-1\).
• The line \(y = 2x-4\) has a slope of \(2\) (positive, steeper slope) and a \(y\) - intercept of \(-4\).

Step3: Analyze the shading regions

• For \(y\leq\frac{1}{3}x - 1\), we shade below the line with slope \(\frac{1}{3}\).
• For \(y\geq2x - 4\), we shade above the line with slope \(2\).
• The solution to the system is the region where both shadings overlap.

Now, let's analyze the graphs:

• Graph W: The shading and line slopes do not match our derived inequalities.
• Graph X: The slopes of the lines (one with gentle positive slope, one with steeper positive slope) and the overlapping shading region (shaded where below the gentle - sloped line and above the steeper - sloped line) match our requirements.
• Graph Y: The shading and line slopes do not match.
• Graph Z: The shading and line slopes do not match.

Answer:

B. Graph X