Question

1. $y \geq x + 4$

Explanation:

Step1: Identify the boundary line

The inequality is \( y \geq x + 4 \). First, consider the boundary line \( y = x + 4 \). This is a linear equation in slope - intercept form (\( y=mx + b \)), where the slope \( m = 1 \) and the y - intercept \( b = 4 \). Since the inequality is \( \geq \), the boundary line will be a solid line (because the points on the line are included in the solution set).

Step2: Determine the region to shade

To find which side of the line to shade, we can use a test point. A common test point is the origin \((0,0)\) (as long as the line does not pass through it). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0\geq0 + 4\), which simplifies to \( 0\geq4 \). This is a false statement. So, the origin is not in the solution set. Therefore, we shade the region that does not contain the origin. In other words, we shade the region above the line \( y=x + 4 \) (since the inequality is \( y\geq x + 4 \), and the origin is below the line, we shade above the line).

Answer:

To graph \( y\geq x + 4 \):

1. Draw the line \( y=x + 4 \) as a solid line (because of the "greater than or equal to" symbol). The line has a slope of 1 and a y - intercept at \( (0,4) \).
2. Shade the region above the line \( y = x+4 \) (since the test point \((0,0)\) does not satisfy the inequality, we shade the opposite side of the line from the origin).