Question

use add work or your scrap paper: provide the steps you used to arrive at the solution these terms: x- and y-intercepts, slope, solid or dashed line, and shade above or shade significance of the shaded area. graph: y - 4x ≤ -2 question help: video message instructor add work calculator

Explanation:

Step1: Rewrite the inequality

Rewrite \( y - 4x \leq -2 \) in slope - intercept form (\( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept). Add \( 4x \) to both sides of the inequality: \( y\leq4x - 2 \).

Step2: Find the y - intercept

For the equation \( y = 4x-2 \), when \( x = 0 \), \( y=-2 \). So the y - intercept is \( (0,-2) \).

Step3: Find the x - intercept

Set \( y = 0 \) in the equation \( y = 4x - 2 \). Then \( 0=4x-2 \). Add 2 to both sides: \( 4x=2 \), and divide by 4: \( x=\frac{2}{4}=\frac{1}{2} \). So the x - intercept is \( (\frac{1}{2},0) \).

Step4: Determine the type of line

Since the inequality is \( y\leq4x - 2 \) (the symbol is \( \leq \), not \( < \)), the line should be solid. This is because the points on the line \( y = 4x-2 \) are included in the solution set.

Step5: Determine the shading direction

To determine where to shade, we can use a test point. A common test point is \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y\leq4x - 2 \): \( 0\leq4(0)-2 \), which simplifies to \( 0\leq - 2 \). This is false. So we shade the region that does not include the test point \( (0,0) \). Since the line is \( y = 4x - 2 \), and the test point \( (0,0) \) is above the line (because when \( x = 0 \), the line has \( y=-2 \) and \( 0>-2 \)), we shade below the line.

Step6: Graph the line and shade

Plot the y - intercept \( (0,-2) \) and the x - intercept \( (\frac{1}{2},0) \). Draw a solid line through these two points (using the slope \( m = 4 \), which means for every 1 unit we move to the right along the x - axis, we move up 4 units along the y - axis). Then shade the region below the solid line. The shaded area represents all the points \( (x,y) \) that satisfy the inequality \( y - 4x\leq - 2 \).

Answer:

To graph \( y - 4x\leq - 2 \):

1. Rewrite as \( y\leq4x - 2 \).
2. Y - intercept: \( (0,-2) \), X - intercept: \( (\frac{1}{2},0) \).
3. Draw a solid line through these points (slope \( m = 4 \)).
4. Shade below the solid line (since \( (0,0) \) does not satisfy \( y\leq4x - 2 \)). The shaded area contains all solutions \((x,y)\) to the inequality.