Question

question 34
which graph is correct for the inequality y < x + 2?
select one

Explanation:

Step1: Analyze the inequality \( y < x + 2 \)

First, consider the boundary line \( y = x + 2 \). The slope of this line is \( 1 \) (since it's in the form \( y = mx + b \) with \( m = 1 \)) and the y - intercept \( b = 2 \). Since the inequality is \( y < x + 2 \), the boundary line should be dashed (because the inequality is strict, \( < \), not \( \leq \)) and we shade the region below the line.

Step2: Find the y - intercept and slope

The y - intercept of \( y = x + 2 \) is \( (0, 2) \), which means the line crosses the y - axis at \( y = 2 \). The slope \( m = 1 \) means for every 1 unit we move to the right (increase in x by 1), we move up 1 unit (increase in y by 1).

Step3: Analyze the shading and line type

• For the inequality \( y < x + 2 \), we shade below the line \( y = x + 2 \). Also, since the inequality is strict (\( < \)), the line should be dashed (not solid).
• Let's check the y - intercept: the line should pass through \( (0, 2) \). Now, let's look at the options. The correct graph should have a dashed line (since \( y < x + 2 \), not \( y\leq x + 2 \)) and the shaded region below the line \( y = x + 2 \).

Looking at the options, the third graph (assuming the order from top to bottom: first, second, third, fourth) has the line with slope 1, y - intercept 2, dashed line (or appropriate line type for strict inequality) and shading below the line. Wait, let's re - evaluate. Wait, maybe the third graph (the one with the purple region below the line that has a y - intercept around 2). Wait, actually, let's check the y - intercept. The line \( y=x + 2 \) passes through \( (0,2) \) and \( (- 2,0) \) (when \( y = 0 \), \( 0=x + 2\Rightarrow x=-2 \)). So the line should go through \( (0,2) \) and \( (-2,0) \). The region below this line (where \( y < x + 2 \)) should be shaded. Also, since the inequality is strict, the line is dashed.

Among the given graphs, the third graph (the one with the purple region that is below the line with slope 1 and y - intercept 2) is the correct one. Wait, maybe the third option (the third graph in the list) has the correct line and shading. Let's assume the graphs are ordered as:

1. First graph: Maybe wrong line or shading.

1. Second graph: Wrong y - intercept (maybe y - intercept at - 4? No, our line has y - intercept 2).

1. Third graph: The line passes through \( (0,2) \) (since when x = 0, y = 2) and \( (-2,0) \) (when y = 0, x=-2), and the shading is below the line, and the line is dashed (or the correct type for strict inequality).

1. Fourth graph: Maybe wrong line.

So the correct graph is the third one (the third option in the list of graphs).

Answer:

The third graph (assuming the graphs are listed from top to bottom as first, second, third, fourth)