Question

1. graph the following system of inequalities (hint: what happens when you multiply or divide by a negative?)?

y > -x + 2
-y ≥ -2x + 1

Explanation:

Step1: Analyze the first inequality

The first inequality is \( y > -x + 2 \). This is a linear inequality in slope - intercept form \( y=mx + b \), where the slope \( m=- 1 \) and the y - intercept \( b = 2 \). Since the inequality is \( y>-x + 2 \), we will draw a dashed line (because the inequality is strict, \(>\) not \(\geq\)) with a slope of - 1 and y - intercept 2, and shade the region above the line.

Step2: Analyze the second inequality

The second inequality is \( -y\geq-2x + 1 \). To get it in slope - intercept form, we multiply both sides by - 1. When we multiply or divide an inequality by a negative number, the direction of the inequality sign changes. So we get \( y\leq2x - 1 \). Here, the slope \( m = 2 \) and the y - intercept \( b=-1 \). Since the inequality is \( y\leq2x - 1 \), we will draw a solid line (because the inequality is non - strict, \(\leq\)) with a slope of 2 and y - intercept - 1, and shade the region below the line.

Step3: Graph the lines and find the solution region

1. For \( y > -x + 2 \):

• Plot the y - intercept at (0, 2). Using the slope \( m=-1=\frac{-1}{1}\), from the point (0, 2), we can go down 1 unit and right 1 unit (or up 1 unit and left 1 unit) to find another point on the line. Since the inequality is \( y > -x+2 \), we draw a dashed line and shade the area above the line.

1. For \( y\leq2x - 1 \):

• Plot the y - intercept at (0, - 1). Using the slope \( m = 2=\frac{2}{1}\), from the point (0, - 1), we can go up 2 units and right 1 unit (or down 2 units and left 1 unit) to find another point on the line. Since the inequality is \( y\leq2x - 1 \), we draw a solid line and shade the area below the line.

The solution to the system of inequalities is the region where the two shaded regions overlap.

Answer:

To graph the system:

1. Graph \( y > -x + 2 \) as a dashed line with slope - 1, y - intercept 2, shade above.
2. Graph \( y\leq2x - 1 \) (after multiplying \( -y\geq-2x + 1 \) by - 1 and reversing the inequality) as a solid line with slope 2, y - intercept - 1, shade below.

The overlapping region of the two shaded areas is the solution.