Question

run it back_module 6 test question 7 of 9 (1 point) | question attempt: 2 of graph the inequality. y < -5x + 1

Explanation:

Step 1: Identify the boundary line

The inequality is \( y < -5x + 1 \). First, consider the equation of the boundary line, which is \( y=-5x + 1 \). Since the inequality is \( y < -5x+1 \) (not \( y\leq - 5x + 1 \)), the boundary line should be a dashed line (because the points on the line \( y=-5x + 1 \) do not satisfy the inequality \( y < -5x + 1 \)).

Step 2: Determine the slope and y - intercept of the boundary line

For a linear equation in the form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. In the equation \( y=-5x + 1 \), the slope \( m=-5 \) and the y - intercept \( b = 1 \). To graph the line \( y=-5x + 1 \):

• Start by plotting the y - intercept. The y - intercept is at the point \( (0,1) \) (since when \( x = 0 \), \( y=1 \)).
• Then, use the slope to find another point. The slope \( m=-5=\frac{-5}{1} \), which means from the point \( (0,1) \), we can move down 5 units (because the numerator of the slope is - 5) and to the right 1 unit (because the denominator of the slope is 1). So we get the point \( (0 + 1,1-5)=(1,-4) \). We can also move up 5 units and to the left 1 unit (since a slope of \( - 5=\frac{5}{-1} \)) from \( (0,1) \) to get the point \( (0 - 1,1 + 5)=(-1,6) \). Draw a dashed line through these points to represent the boundary line \( y=-5x + 1 \).

Step 3: Determine the region to shade

To determine which side of the line to shade, we can use a test point. A common test point is \( (0,0) \) (as long as it is not on the boundary line). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y < -5x+1 \):
\( 0< - 5(0)+1 \)
\( 0 < 0 + 1 \)
\( 0<1 \), which is a true statement. So the point \( (0,0) \) satisfies the inequality. Therefore, we shade the region that contains the point \( (0,0) \), which is the region below the dashed line \( y=-5x + 1 \).

Answer:

To graph \( y < -5x + 1 \):

1. Draw the dashed line \( y=-5x + 1 \) (dashed because the inequality is strict, \( y < \), not \( y\leq \)) with slope - 5 and y - intercept at \( (0,1) \).
2. Shade the region below the dashed line (the region containing the test point \( (0,0) \) since \( 0< - 5(0)+1 \) is true).