Question

y ≤ -3x - 1

Explanation:

Step 1: Identify the type of inequality

The inequality is \( y \leq -3x - 1 \), which is a linear inequality. To graph it, we first consider the boundary line \( y = -3x - 1 \).

Step 2: Determine the slope and y-intercept

For the line \( y = -3x - 1 \), the slope \( m = -3 \) and the y-intercept \( b = -1 \). So, the line crosses the y-axis at \( (0, -1) \).

Step 3: Graph the boundary line

Since the inequality is \( \leq \), the boundary line should be solid (because the inequality includes equality). We can find another point using the slope. From \( (0, -1) \), using the slope \( -3 \) (which is \( \frac{-3}{1} \)), we move down 3 units and right 1 unit to get \( (1, -4) \), or up 3 units and left 1 unit to get \( (-1, 2) \). Plot these points and draw a solid line through them.

Step 4: Shade the region

To determine which side of the line to shade, we can test a point not on the line. A common test point is \( (0, 0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality:
\( 0 \leq -3(0) - 1 \)
\( 0 \leq -1 \)
This is false. So, we shade the side of the line that does not include \( (0, 0) \). Since the line has a negative slope, the region below the line (where \( y \) values are less than or equal to \( -3x - 1 \)) should be shaded.

Answer:

To graph \( y \leq -3x - 1 \):

1. Draw the solid line \( y = -3x - 1 \) (passing through \( (0, -1) \) and other points like \( (1, -4) \) or \( (-1, 2) \)).
2. Shade the region below the solid line (since the test point \( (0, 0) \) does not satisfy the inequality, we shade the opposite side of the line from \( (0, 0) \)).