Question

Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions:

Answer 1

The given inequalities are
$y \ \textless \ 4x-8 \\\\ y \ge \frac{5}{2x+5}$

Part A
The graph is shown below.
The shaded regions indicate where both inequalities hold.

Notice that when y changes sign, the sign of the inequality gets flipped.
For example,
y < 4x - 8 becomes  - y > -4x + 8.

Part B
Visually, the point (5, -8) is not included in the solution area.

Test the inequality y < 4x - 8
When x = 5, we should be in the shaded area.
y < 4*8 - 8 
y < 24      
y< 24 is conditionally true.
Notice that y is below the red curve, which approaches 0 asymptotically from above.

Answer:
The point (5, -8) is not in the solution area.

Answer 2

Answer:

Part A: The graph of given system of inequities is shown below.

Part B: (5, −8) is not included in the solution area for the system.

Step-by-step explanation:

The given inequalities are

$y$

$y\ge -\frac{5}{2}x+5$

Part A:

The related equations for the given inequalities are

$y=4x-8$

$y=-\frac{5}{2}x+5$

Slope intercept form of a line is

$y=mx+b$

where, m is slope and b is y-intercept.

On comparing the related equation with slope intercept form, we get

The slope of line 1 is 4 and y-intercept is -8.

The slope of line 2 is -5/2 and y-intercept is 5.

Table of values are

Line 1                   Line 2

x     y                   x        y

0    -8                  0       5

2     0                  2      0

Using these order pairs plot both line on a coordinate plane.

First line is dotted because the sign of inequality is <.

second line is dotted because the sign of inequality is ≥.

Check both inequalities by (0,0).

$(0)$ False

$(0)\ge -\frac{5}{2}(0)+5\Rightarrow 0\ge 5$ False

It means (0,0) is not included in the shaded region of any line.

The graph of given system of inequities is shown below.

Part B:

We need to check whether (5,-8) is included in the solution area for the system.

Substitute x=5 and y=-8 in inequality (1).

$(-8)$

$-8$

$-8$

This statement is true. It means (5,-8) is included in the shaded area of inequality (1).

Substitute x=5 and y=-8 in inequality (2).

$-8\ge -\frac{5}{2}(5)+5$

$-8\ge -7.5$

This statement is false. It means (5,-8) is not included in the shaded area of inequality (2).

Therefore, (5, −8) is not included in the solution area for the system.