Question

on a piece of paper, graph ( y geq x - 1 ). then determine which answer choice matches the graph you drew.

a

b

c

d

text description for graph

a. graph a

b. graph d

c. graph b

d. graph c

Explanation:

Step1: Analyze the inequality

The inequality is \( y \geq x - 1 \). First, we consider the boundary line \( y = x - 1 \). The slope of this line is \( 1 \) and the y - intercept is \( - 1 \), so it passes through \( (0,-1) \) and for \( x = 3 \), \( y=3 - 1=2 \), so it also passes through \( (3,2) \). Since the inequality is \( y\geq x - 1 \), the boundary line should be solid (because the inequality includes equality, \( \geq \) means the line is part of the solution set).

Step2: Determine the shading region

To find the correct shading, we can test a point. Let's use the origin \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y\geq x - 1 \): \( 0\geq0 - 1 \), which simplifies to \( 0\geq - 1 \), and this is true. So the region that includes the origin should be shaded.

Now let's analyze each graph:

• Graph A: The line is solid. Let's check the shading. The origin \( (0,0) \) is in the shaded region (since the shaded area is above the line and includes the origin - like region). Let's verify the line: passes through \( (0,-1) \) and \( (3,2) \), solid line.
• Graph B: The shaded region is below the line. Testing \( (0,0) \) in \( y\geq x - 1 \), \( 0\geq - 1 \) is true, but the shaded region here is below the line, which would correspond to \( y\leq x - 1 \), so this is incorrect.
• Graph C: The line is dashed, but our boundary line should be solid (because of \( \geq \)), so this is incorrect.
• Graph D: The line is dashed, and the shading is not correct. Since the line should be solid, this is incorrect.

Answer:

A. Graph A