Question

As a computer technician, andre makes $20 per hour to diagnose a problem and $25 per hour to fix a problem. he works fewer than 10 hours per week, but wants to make at least $200 per week. the inequalities 20x 25y ≥ 200 and

Answer 1

The question deals with real-world applications of linear equations and inequalities to determine earnings and make economic decisions. For Andre, correct inequalities need to be established to ensure his weekly earnings. For the consultant, it's more economical to work at her high hourly rate and buy vegetables rather than grow them.

Step-by-step explanation:

The subject of the student's question is Mathematics, and it pertains to linear equations and inequalities. Specifically, we are looking at a real-world application involving a computer technician named Andre and his earnings, which consist of a $20 hourly rate for diagnosing problems, and a $25 hourly rate for fixing problems. The student has presented two separate but related parts to address:

1. The determination of the inequalities that represent Andre's goal to make at least $200 per week, given his hourly rates and the condition that he works fewer than 10 hours per week.
2. An example involving a consultant who earns $200 per hour to show why it makes more economic sense to work and buy vegetables as opposed to growing them herself.

For the first scenario, if we let x represent the hours spent diagnosing and y represent the hours spent fixing, we should find the correct inequalities to ensure Andre makes at least $200 per week. The unfinished given inequality is incomplete and requires correction to fully address the question. It appears to be two separate inequalities: 20x + 25y ≥ 200, ensuring he makes at least $200, and another inequality that represents his work hours, likely x + y < 10 since he works fewer than 10 hours per week.

For the second scenario, it is more economically sensible for the consultant to focus on her consulting job because, at a rate of $200 per hour, the opportunity cost of her time is very high. Every hour spent on a less productive activity, like attempting to grow vegetables poorly, is an hour not spent earning $200. Therefore, it makes better financial sense to work the high-paying job and use a portion of those earnings to buy vegetables.

Answer 2

The correct options are b, c, and e.

The correct statements are that the line
`\( x + y \leq 10 \)` has a negative slope and a positive y-intercept, the line
`\( 20x + 25y \geq 200 \)` is solid with the region above it shaded, and the overlapping region of the inequalities includes the point (4, 5).

To answer the question , we need to analyze the given inequalities and determine the characteristics of their graphs. Here are the inequalities:

1.
`\( 20x + 25y \geq 200 \)`

2.
`\( x + y \leq 10 \)`

Let's break down what each inequality represents and then address the statements one by one.

Inequality 1:
`\( 20x + 25y \geq 200 \)`

This inequality represents Andre's goal to make at least $200 per week. The 20x term represents the $20 per hour for diagnosing a problem, and the 25y term represents the $25 per hour for fixing a problem.

The graph of this inequality will be a line that divides the plane into two regions: one where the inequality is satisfied (above the line), and one where it is not (below the line).

Since the inequality is greater than or equal to, the line itself will be solid, and the region above the line will be shaded.

Slope and y-intercept:

The slope of the line can be found by rearranging the equation into slope-intercept form
`\( y = mx + b \)`, where m is the slope and b is the y-intercept.

Let's rearrange it to find the slope and y-intercept:

`\[ 25y \geq -20x + 200 \]`

`\[ y \geq -(20)/(25)x + 8 \]`

`\[ y \geq -(4)/(5)x + 8 \]`

Here, the slope
`\( m = -(4)/(5) \)` which is negative, and the y-intercept b = 8 which is positive.

Inequality 2:
`\( x + y \leq 10 \)`

This inequality represents the condition that Andre works fewer than 10 hours per week. The graph of this inequality will also be a line. Since the inequality is less than or equal to, the region below the line will be shaded, and the line itself will be solid.

Slope and y-intercept:

By rearranging this equation into the slope-intercept form, we get:

`\[ y \leq -x + 10 \]`

Here, the slope m = -1 which is negative, and the y-intercept b = 10 which is positive.

Now let's address each statement based on our analysis:

• The line
  `\( 20x + 25y \geq 200 \)` has a positive slope and a negative y-intercept. (False, as we've found a negative slope and a positive y-intercept)
• The line
  `\( x + y \leq 10 \)` has a negative slope and a positive y-intercept. (True, slope is -1 and y-intercept is 10)
• The line representing
  `\( 20x + 25y \geq 200 \)` is solid and the graph is shaded above the line. (True, it is solid and shaded above)
• The line representing
  `\( x + y \leq 10 \)` is dashed and the graph is shaded above the line. (False, it is solid and shaded below)

The overlapping region contains the point (4, 5). To verify this, we can plug in the values into both inequalities.

For
`\( 20x + 25y \geq 200 \)`:

`\[ 20(4) + 25(5) \geq 200 \]`

`\[ 80 + 125 \geq 200 \]`

`\[ 205 \geq 200 \]` (True)

For
`\( x + y \leq 10 \)`:

`\[ 4 + 5 \leq 10 \]`

`\[ 9 \leq 10 \]` (True)

Hence, the overlapping region does contain the point (4, 5). (True)

So, the correct statements are:

• The line
  `\( x + y \leq 10 \)` has a negative slope and a positive y-intercept.
• The line representing
  `\( 20x + 25y \geq 200 \)` is solid and the graph is shaded above the line.
• The overlapping region contains the point (4, 5).

The complete question is here:

As a computer technician, Andre makes
`$\$ 20$` per hour to diagnose a problem and
`$\$ 25$` per hour to fix a problem. He works fewer than 10 hours per week, but wants to make at least
`$\$ 200$` per week. The inequalities
`$20 x+25 y \geq 200$` and
`$x+y < 10$` represent the situation. Which is true of the graph of the solution set? Check all that apply.

a. The line
`$20 x+25 y \geq 200$` has a positive slope and a negative y-intercept.

b. The line
`$x+y < 10$` has a negative slope and a positive y-intercept.

c. The line representing
`$20 x+25 y \geq 200$` is solid and the graph is shaded above the line.

d. The line representing x+y<10 is dashed and the graph is shaded above the line.

e. The overlapping region contains the point (4,5).