Question

1. $y \leq -7x + 4$

Explanation:

Step1: Identify the boundary line

The inequality is \( y \leq -7x + 4 \). First, we consider the boundary line \( y = -7x + 4 \). This is a linear equation in slope - intercept form (\( y=mx + b \)), where the slope \( m=-7 \) and the y - intercept \( b = 4 \). To graph the line, we start by plotting the y - intercept. The y - intercept is at \( (0,4) \) (since when \( x = 0 \), \( y=-7(0)+4 = 4 \)). Then, using the slope, for a slope of \( - 7=\frac{-7}{1} \), from the point \( (0,4) \), we can go down 7 units and right 1 unit to get another point on the line, or up 7 units and left 1 unit. Since the inequality is \( y\leq - 7x + 4 \), the boundary line should be a solid line (because the inequality includes the equal sign).

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: \( 0\leq-7(0)+4 \), which simplifies to \( 0\leq4 \). This statement is true. So, we shade the region that includes the origin.

Graphing Steps Summary:

1. Plot the y - intercept \( (0,4) \).
2. Use the slope \( - 7 \) to find another point on the line \( y=-7x + 4 \) (e.g., from \( (0,4) \), moving to \( (1, - 3) \) since \( y=-7(1)+4=-3 \)).
3. Draw a solid line through these points.
4. Test the point \( (0,0) \): since \( 0\leq4 \) is true, shade the region that contains \( (0,0) \) (the region below the line \( y = - 7x+4 \)).

(Note: Since the question is about graphing the inequality, the final answer is the graph with the solid line \( y=-7x + 4 \) and the region below the line shaded.)

Answer:

Step1: Identify the boundary line

The inequality is \( y \leq -7x + 4 \). First, we consider the boundary line \( y = -7x + 4 \). This is a linear equation in slope - intercept form (\( y=mx + b \)), where the slope \( m=-7 \) and the y - intercept \( b = 4 \). To graph the line, we start by plotting the y - intercept. The y - intercept is at \( (0,4) \) (since when \( x = 0 \), \( y=-7(0)+4 = 4 \)). Then, using the slope, for a slope of \( - 7=\frac{-7}{1} \), from the point \( (0,4) \), we can go down 7 units and right 1 unit to get another point on the line, or up 7 units and left 1 unit. Since the inequality is \( y\leq - 7x + 4 \), the boundary line should be a solid line (because the inequality includes the equal sign).

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: \( 0\leq-7(0)+4 \), which simplifies to \( 0\leq4 \). This statement is true. So, we shade the region that includes the origin.

Graphing Steps Summary:

1. Plot the y - intercept \( (0,4) \).
2. Use the slope \( - 7 \) to find another point on the line \( y=-7x + 4 \) (e.g., from \( (0,4) \), moving to \( (1, - 3) \) since \( y=-7(1)+4=-3 \)).
3. Draw a solid line through these points.
4. Test the point \( (0,0) \): since \( 0\leq4 \) is true, shade the region that contains \( (0,0) \) (the region below the line \( y = - 7x+4 \)).

(Note: Since the question is about graphing the inequality, the final answer is the graph with the solid line \( y=-7x + 4 \) and the region below the line shaded.)