Question

graph the system of linear inequalities and shade in the solution set. if there are no solutions, graph the corresponding lines and do not shade in any region.
$x - y > 1$
$y < -\frac{1}{2}x + 5$

Explanation:

Step1: Analyze the first inequality \(x - y > 1\)

Rewrite it in slope - intercept form (\(y=mx + b\)): \(y\) or \(<\)) and we shade below the line.

Step2: Analyze the second inequality \(y<-\frac{1}{2}x + 5\)

The boundary line is \(y =-\frac{1}{2}x+5\), with a slope of \(-\frac{1}{2}\) and a \(y\) - intercept of \(5\). Since the inequality is \(y<-\frac{1}{2}x + 5\) (strict inequality), the line is dashed and we shade below the line.

Step3: Graph the lines and find the solution set

• For \(y=x - 1\):
• When \(x = 0\), \(y=-1\); when \(y = 0\), \(x = 1\). Plot the points \((0,-1)\) and \((1,0)\) and draw a dashed line through them. Shade the region below this line.
• For \(y=-\frac{1}{2}x + 5\):
• When \(x = 0\), \(y = 5\); when \(y=0\), \(x = 10\). Plot the points \((0,5)\) and \((10,0)\) and draw a dashed line through them. Shade the region below this line.
• The solution set of the system is the region that is shaded by both inequalities. To find the intersection, we can also find the point of intersection of the two boundary lines \(y=x - 1\) and \(y=-\frac{1}{2}x + 5\) by setting \(x - 1=-\frac{1}{2}x + 5\).
• Add \(\frac{1}{2}x\) to both sides: \(x+\frac{1}{2}x-1=5\)
• Combine like terms: \(\frac{3}{2}x=6\)
• Multiply both sides by \(\frac{2}{3}\): \(x = 4\)
• Substitute \(x = 4\) into \(y=x - 1\), we get \(y=4 - 1=3\). So the lines intersect at \((4,3)\).
• The solution region is the area that is below both \(y=x - 1\) and \(y=-\frac{1}{2}x + 5\), bounded by the two dashed lines.

(Note: Since the original graph provided in the problem seems to be incorrect for this system of inequalities, the correct graphing process is as above. If we were to graph it properly, we would have two dashed lines and a shaded region where their "below" regions overlap.)

Answer:

To graph the system:

1. Draw the dashed line \(y = x-1\) (slope \(1\), \(y\) - intercept \(-1\)) and shade below it.
2. Draw the dashed line \(y=-\frac{1}{2}x + 5\) (slope \(-\frac{1}{2}\), \(y\) - intercept \(5\)) and shade below it.
3. The solution set is the overlapping shaded region (bounded by the two dashed lines and below both lines), with the intersection point of the two lines at \((4,3)\).