Question

graph the inequality.
$y \geq -2x + 2$

Explanation:

Step1: Identify the boundary line

The inequality is \( y \geq -2x + 2 \). First, consider the equation of the boundary line, which is \( y = -2x + 2 \). This is a linear equation in slope - intercept form (\( y=mx + b \)), where the slope \( m=-2 \) and the y - intercept \( b = 2 \). To graph the line, we can find two points. When \( x = 0 \), \( y=-2(0)+2=2 \), so the point \( (0,2) \) is on the line. When \( y = 0 \), \( 0=-2x + 2\), solving for \( x \) gives \( 2x=2\) or \( x = 1 \), so the point \( (1,0) \) is on the line. Since the inequality is \( y\geq - 2x + 2 \) (the "greater than or equal to" symbol), the boundary line should be a solid line (because the inequality includes equality).

Step2: Determine the region to shade

To determine which side of the line to shade, we can use a test point. A common test point is \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y\geq -2x + 2 \): \( 0\geq -2(0)+2\), which simplifies to \( 0\geq2 \). This is false. So we shade the region that does not contain the test point \( (0,0) \). In other words, we shade the region above the line \( y=-2x + 2 \) (since the test point below the line did not satisfy the inequality).

Answer:

1. Draw the solid line \( y=-2x + 2 \) (passing through \( (0,2) \) and \( (1,0) \)).
2. Shade the region above the line \( y = - 2x+2 \).