**Graphing a System of Inequalities**On the set of axes below, we will graph the following system of inequalities:\[ 2x + y \ge 8 \]\[ y - 5 < 3x \]To graph these inequalities, follow these steps:### Step-by-Step Instructions:1. **Graphing the Boundary Line for $2x + y \ge 8$:** - Rewrite the inequality in slope-intercept form: $ y \ge -2x + 8 $. - Plot the boundary line $ y = -2x + 8 $. This line has a slope of -2 and a y-intercept of 8. - Since the inequality is $ \ge $, shade the region above the line.2. **Graphing the Boundary Line for $y - 5 < 3x$:** - Rewrite the inequality in slope-intercept form: $ y < 3x + 5 $. - Plot the boundary line $ y = 3x + 5 $. This line has a slope of 3 and a y-intercept of 5. - Since the inequality is $ < $, shade the region below the line with a dashed line to indicate that points on the line are not included in the solution set.3. **Determine the Solution Region:** - The solution region will be the area where the shaded regions of both inequalities overlap.### Determine if the Point (1, 8) is in the Solution Set:Now, let's determine if the point $ (1, 8) $ is in the solution set:1. Substitute $ x = 1 $ and $ y = 8 $ into $ 2x + y \ge 8 $: \[ 2(1) + 8 \ge 8 \implies 2 + 8 \ge 8 \implies 10 \ge 8 \] This inequality is true.2. Substitute $ x = 1 $ and $ y = 8 $ into $ y - 5 < 3x $: \[ 8 - 5 < 3(1) \implies 3 < 3 \] This inequality is false since 3 is not less than 3.Therefore, the point \( (1, 8

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Answered: On the set of axes below, graph the…

Math

Algebra

On the set of axes below, graph the ystem of inequalities: 2x+y 2 8 y-5< 3x

Intermediate Algebra

19th Edition

ISBN:9780998625720

Author:Lynn Marecek

Publisher:Lynn Marecek

Chapter3: Graphs And Functions

Section3.4: Graph Linear Inequalities In Two Variables

Problem 277E: Harrison works two part time jobs. One at a gas station that pays $11 an hour and the other is IT...

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Question

**Graphing a System of Inequalities**

On the set of axes below, we will graph the following system of inequalities:

\[
2x + y \ge 8
\]
\[
y - 5 < 3x
\]

To graph these inequalities, follow these steps:

### Step-by-Step Instructions:

1. **Graphing the Boundary Line for $2x + y \ge 8$:**
- Rewrite the inequality in slope-intercept form: $ y \ge -2x + 8 $.
- Plot the boundary line $ y = -2x + 8 $. This line has a slope of -2 and a y-intercept of 8.
- Since the inequality is $ \ge $, shade the region above the line.

2. **Graphing the Boundary Line for $y - 5 < 3x$:**
- Rewrite the inequality in slope-intercept form: $ y < 3x + 5 $.
- Plot the boundary line $ y = 3x + 5 $. This line has a slope of 3 and a y-intercept of 5.
- Since the inequality is $ < $, shade the region below the line with a dashed line to indicate that points on the line are not included in the solution set.

3. **Determine the Solution Region:**
- The solution region will be the area where the shaded regions of both inequalities overlap.

### Determine if the Point (1, 8) is in the Solution Set:

Now, let's determine if the point $ (1, 8) $ is in the solution set:

1. Substitute $ x = 1 $ and $ y = 8 $ into $ 2x + y \ge 8 $:
\[
2(1) + 8 \ge 8 \implies 2 + 8 \ge 8 \implies 10 \ge 8
\]
This inequality is true.

2. Substitute $ x = 1 $ and $ y = 8 $ into $ y - 5 < 3x $:
\[
8 - 5 < 3(1) \implies 3 < 3
\]
This inequality is false since 3 is not less than 3.

Therefore, the point \( (1, 8

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