Question

1. graph the inequality

$y > 2x - 4$

1. graph the inequality

$5x - 2y \leq 8$

1. which points are solutions to the

inequality below?

1. which ordered pair is not a solution to the system of inequalities?

$3x + y > -3$
$x + 2y < 4$
a. $(1, 1)$
b. $(-1, 2)$
c. $(4, -3)$
d. $(-5, 1)$

Explanation:

Problem 17: Graph \( y > 2x - 4 \)

Step 1: Identify the boundary line

The inequality is \( y > 2x - 4 \). The boundary line is \( y = 2x - 4 \), which is a straight line with slope \( 2 \) and y-intercept \( -4 \). Since the inequality is \( > \) (not \( \geq \)), the line should be dashed.

Step 2: Determine the region to shade

To find which side of the line to shade, we can test a point not on the line. Let's use the origin \( (0,0) \):
Substitute \( x = 0 \), \( y = 0 \) into \( y > 2x - 4 \):
\( 0 > 2(0) - 4 \)
\( 0 > -4 \), which is true. So we shade the region that includes the origin.

Step 3: Graph the line and shade

• Plot the y-intercept at \( (0, -4) \).
• Use the slope \( 2 \) (rise 2, run 1) to find another point, e.g., \( (1, -2) \).
• Draw a dashed line through these points.
• Shade the region above the line (since the origin is in that region and satisfies the inequality).

Problem 18: Graph \( 5x - 2y \leq 8 \)

Step 1: Rewrite in slope-intercept form

Solve \( 5x - 2y \leq 8 \) for \( y \):
\( -2y \leq -5x + 8 \)
Divide both sides by \( -2 \) (remember to reverse the inequality sign):
\( y \geq \frac{5}{2}x - 4 \)
The boundary line is \( y = \frac{5}{2}x - 4 \), with slope \( \frac{5}{2} \) and y-intercept \( -4 \). Since the inequality is \( \geq \), the line is solid.

Step 2: Determine the region to shade

Test the origin \( (0,0) \) in \( y \geq \frac{5}{2}x - 4 \):
\( 0 \geq \frac{5}{2}(0) - 4 \)
\( 0 \geq -4 \), which is true. So we shade the region that includes the origin.

Step 3: Graph the line and shade

• Plot the y-intercept at \( (0, -4) \).
• Use the slope \( \frac{5}{2} \) (rise 5, run 2) to find another point, e.g., \( (2, 1) \) (since \( \frac{5}{2}(2) - 4 = 5 - 4 = 1 \)).
• Draw a solid line through these points.
• Shade the region above the line (since the origin is in that region and satisfies the inequality).

Problem 19: Identify solution points (assuming the inequality is \( y \geq x - 2 \) from the graph)

First, determine the inequality from the graph. The line has a slope of \( 1 \) and y-intercept \( -2 \), so the inequality is \( y \geq x - 2 \) (solid line, shaded above). Now test each point:

• \( (0, -4) \): \( -4 \geq 0 - 2 \)? \( -4 \geq -2 \)? No.
• \( (2, 4) \): \( 4 \geq 2 - 2 \)? \( 4 \geq 0 \)? Yes.
• \( (4, -3) \): \( -3 \geq 4 - 2 \)? \( -3 \geq 2 \)? No.
• \( (3, 2) \): \( 2 \geq 3 - 2 \)? \( 2 \geq 1 \)? Yes.
• \( (5, 4) \): \( 4 \geq 5 - 2 \)? \( 4 \geq 3 \)? Yes.
• \( (-3, -5) \): \( -5 \geq -3 - 2 \)? \( -5 \geq -5 \)? Yes (since it's \( \geq \)).

So the solution points are \( (2, 4) \), \( (3, 2) \), \( (5, 4) \), \( (-3, -5) \).

Problem 20: Find the non - solution to \(

$$\begin{cases}3x + y > - 3\\x + 2y < 4\end{cases}$$

\)

Answer:

s:

1. Graph with dashed line \( y = 2x - 4 \) and shade above.
2. Graph with solid line \( y=\frac{5}{2}x - 4 \) and shade above.
3. Solution points: \( (2, 4) \), \( (3, 2) \), \( (5, 4) \), \( (-3, -5) \)
4. D. \( (-5, 1) \)