Question

graph the inequality. y > -5x + 1

Explanation:

Step1: Analyze the inequality form

The inequality is \( y > -5x + 1 \), which is a linear inequality in two variables. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For \( y=-5x + 1 \), the slope \( m=-5 \) and the y - intercept \( b = 1 \).

Step2: Determine the line type

Since the inequality is \( y > -5x+1 \) (not \( y\geq - 5x + 1 \)), the boundary line should be a dashed line. We plot the line \( y=-5x + 1 \) first. To plot the line, we can use the y - intercept: when \( x = 0 \), \( y=1 \), so the point \( (0,1) \) is on the line. We can also find another point by using the slope. The slope \( m=-5=\frac{-5}{1} \), so from the point \( (0,1) \), we can move 1 unit to the right (increase \( x \) by 1) and 5 units down (decrease \( y \) by 5) to get the point \( (1,1 - 5)=(1,-4) \).

Step3: Determine the shading region

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 line). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y>-5x + 1 \): \( 0>-5(0)+1\Rightarrow0 > 1 \), which is false. So we shade the region that does not contain the point \( (0,0) \). In other words, we shade the region above the line \( y=-5x + 1 \) (since the inequality is \( y \) greater than the linear expression, we shade above the line for a line with a negative slope? Wait, no, the direction of shading depends on the test point. Since \( (0,0) \) gives a false statement, we shade the opposite side of the line from \( (0,0) \).

Step4: Check the graph elements

• The boundary line: Since the inequality is strict (\(>\)), the line should be dashed.
• The slope and intercept: The line \( y=-5x + 1 \) has a y - intercept at \( (0,1) \) and a slope of - 5, which matches the equation.
• The shading: The region above the dashed line \( y =-5x+1 \) should be shaded.

Answer:

To graph the inequality \( y > - 5x+1 \):

1. Draw a dashed line for \( y=-5x + 1 \) (using the y - intercept \( (0,1) \) and the slope \( - 5 \) to plot the line).
2. Shade the region above the dashed line (determined by testing a point like \( (0,0) \) which does not satisfy the inequality, so we shade the opposite region).