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A class trip to a beach has been planned for your senior trip. The lifeguard is only on duty when the temperature is between 70 degrees and 105 degrees. There is room for 60 people on your trip. Write the constraints to represent this real-world problem, where x is the temperature and y is the number of people on your trip.

x > 70 and y < 105
0 < x ≤ 60 and 70 < y < 105
x < 105 and y > 70
70 < x < 105 and 0 < y ≤ 60
Question 2(Multiple Choice Worth 1 points)
(05.06 LC)

Which of the following points lie in the solution set to the following system of inequalities?

y ≤ x − 5
y ≥ −x − 4

(−5, 2)
(5, −2)
(−5, −2)
(5, 2)
Question 3(Multiple Choice Worth 1 points)
(05.06 MC)

A company dyes two sizes of rugs. A small rug requires 8 hours for dyeing, and a medium-size rug requires 12 hours for dyeing. The dyers have to make at least 22 rugs, and they must do it in less than 240 hours. Let x equal small rugs and y equal medium rugs. Which of the following inequalities can be paired with x + y ≥ 22 to create a system that represents this situation?

8x + 12y > 240
12x + 8y > 240
8x + 12y < 240
12x + 8y < 240
Question 4(Multiple Choice Worth 1 points)
(05.06 LC)

Choose the graph that represents the following system of inequalities:

y ≤ −3x + 1
y ≤ 1 over 2x + 3

In each graph, the area for f(x) is shaded and labeled A, the area for g(x) is shaded and labeled B, and the area where they have shading in common is labeled AB.

Graph of two intersecting lines. Both lines are solid. One line f of x passes through points negative 2, 2 and 0, 3 and is shaded above the line. The other line g of x passes through points 0, 1 and 1, negative 2 and is shaded above the line.
Graph of two lines intersecting lines. Both lines are solid. One line g of x passes through points negative 2, 2 and 0, 3 and is shaded below the line. The other line f of x passes through points 0, 1 and 1, negative 2 and is shaded above the line.
Graph of two intersecting lines. Both lines are solid. One line passes g of x through points negative 2, 2 and 0, 3 and is shaded below the line. The other line f of x passes through points 0, 1 and 1, negative 2 and is shaded below the line.
Graph of two intersecting lines. Both lines are solid. One line f of x passes through points negative 2, 2 and 0, 3 and is shaded above the line. The other line f of x passes through points 0, 1 and 1, negative 2 and is shaded below the line.
Question 5(Multiple Choice Worth 1 points)
(05.06 MC)

Solve the following systems of inequalities and select the correct graph:

2x − y < 4
x + y < −1

In each graph, the area for f(x) is shaded and labeled A, the area for g(x) is shaded and labeled B, and the area where they have shading in common is labeled AB.

a dashed line g of x rising left to right that is shaded below and a dashed line f of x that is falling left to right that is shaded above
a dashed line f of x rising left to right that is shaded above and a dashed line that is falling g of x left to right that is shaded above
a dashed line g of x rising left to right that is shaded above and a dashed line f of x that is falling left to right that is shaded below
a dashed line g of x rising left to right that is shaded below and a dashed line f of x that is falling left to right that is shaded below
Question 6(Multiple Choice Worth 1 points)
(05.06 MC)

The coordinate grid shows points A through K. What point is a solution to the system of inequalities?

y < −2x + 10
y < 1 over 2x − 2

coordinate grid with plotted ordered pairs, point A at negative 5, 4 point B at 4, 7 point C at negative 2, 7 point D at negative 7, 1 point E at 4, negative 2 point F at 1, negative 6 point G at negative 3, negative 10 point H at negative 4, negative 4 point I at 9, 3 point J at 7, negative 4 and point K at 2, 3

I
B
A
F
Question 7(Multiple Choice Worth 1 points)
(05.06 MC)

In the graph, the area below f(x) is shaded and labeled A, the area below g(x) is shaded and labeled B, and the area where f(x) and g(x) have shading in common is labeled AB.

Graph of two intersecting lines. One line g of x is solid and goes through the points negative 3, 0, negative 4, negative 1 and is shaded in below the line. The other line f of x is solid, and goes through the points 1, 1, 2, negative 1 and is shaded in below the line.

The graph represents which system of inequalities?

y ≤ −2x + 3
y ≤ x + 3
y ≥ −2x + 3
y ≥ x + 3
y ≤ −3x + 2
y ≤ −x + 2
y > −2x + 3
y > x + 3
Question 8(Multiple Choice Worth 1 points)
(05.06 MC)

A bicycle manufacturing company makes a particular type of bike. Each child bike requires 4 hours to build and 4 hours to test. Each adult bike requires 6 hours to build and 4 hours to test. With the number of workers, the company is able to have up to 120 hours of building time and 100 hours of testing time for a week. If c represents child bikes and a represents adult bikes, can the company build 5 child bikes and 15 adult bikes in a week.

No, because the bike order does not meet the restrictions of 4c + 6a ≤ 120 and 4c + 4a ≤ 100
No, because the bike order does not meet the restrictions of 4c + 4a ≤ 120 and 6c + 4a ≤ 100
Yes, because the bike order meets the restrictions of 4c + 6a ≤ 120 and 4c + 4a ≤ 100
Yes, because the bike order meets the restrictions of 4c + 4a ≤ 120 and 6c + 4a ≤ 100
Question 9(Multiple Choice Worth 1 points)
(05.06 MC)

In the graph, the area above f(x) is shaded and labeled A, the area below g(x) is shaded and labeled B, and the area where f(x) and g(x) have shading in common is labeled AB.

Graph of two intersecting lines. The line g of x is solid, and goes through the points 0, negative 2 and 1, 0 and is shaded in below the line. The other line f of x is dashed, and goes through the points 0, 3, 3, 0 and is shaded in above the line.

The graph represents which system of inequalities?

y > 2x − 3
y > −x − 3
y < 2x − 2
y < −x + 3
y ≤ 2x − 2
y > −x + 3
None of the above
Question 10(Multiple Choice Worth 1 points)
(05.06 MC)

A summer camp cookout is planned for the campers and their families. There is room for 200 people. Each adult costs $4, and each camper costs $3. There is a maximum budget of $750. Write the system of inequalities to represent this real-world scenario, where x is the number of adults and y is the number of campers.

x + y ≤ 200
4x + 3y ≤ 750
x + y ≤ 750
4x + 3y ≤ 200
x + y ≤ 200
3x + 4y ≤ 750
x + y ≤ 750
3x + 4y ≤ 200

1 answer

Here are the answers to each of your questions:

Question 1

To represent the constraints for the class trip to the beach: Correct answer: 70 < x < 105 and 0 < y ≤ 60

Question 2

We need to find points that satisfy the system of inequalities:

  1. \( y \leq x - 5 \)
  2. \( y \geq -x - 4 \)
  • Check the points:
    • (−5, 2): Not valid for \( y \leq x - 5 \) (2 is not ≤ -10).
    • (5, −2): Valid.
    • (−5, −2): Valid.
    • (5, 2): Valid.

The points (5, -2) and (−5, −2) are candidates. However, (5, 2) is NOT valid.

Correct answer: (5, −2)

Question 3

The system needs to create a valid inequality to represent the dyeing hours constraint. To pair with \( x + y \geq 22 \): Correct answer: 8x + 12y < 240

Question 4

For the inequalities:

  1. \( y \leq -3x + 1 \) (shaded below)
  2. \( y \leq \frac{1}{2}x + 3 \) (shaded below)

We look for a graph where both areas are shaded below their respective lines. Therefore, we need the shading to overlap below both lines.

Correct answer: The last option (f below, g below).

Question 5

For the system of inequalities:

  1. \( 2x - y < 4 \) → \( y > 2x - 4 \) (shaded above)
  2. \( x + y < -1 \) → \( y < -x - 1 \) (shaded below)

Look for the graph with the areas of f shaded above and g shaded below.

Correct answer: A dashed line f of x rising left to right that is shaded above, and a dashed line g of x that is falling left to right that is shaded below.

Question 6

To check for points that satisfy:

  1. \( y < -2x + 10 \)
  2. \( y < \frac{1}{2}x - 2 \)

After plotting each point against both inequalities, Point I (9, 3) is a valid candidate if it fits.

Correct answer: I

Question 7

Given shading below two solid lines:

  1. One line going through points (0, 2) and (1, 0), likely representing \( y \leq -2x + 3 \).
  2. The other line through points (3, 0) and (0, 2), suggesting \( y \leq x + 3 \).

Correct answer: y ≤ −2x + 3 and y ≤ x + 3

Question 8

To check if the company can build:

  • 5 child bikes and 15 adult bikes. Check if:
  1. \( 4c + 6a \leq 120 \) (checking building hours).
  2. \( 4c + 4a \leq 100 \) (checking testing hours).

Substituting \( c = 5 \) and \( a = 15 \):

  1. \( 4(5) + 6(15) > 120 \) (Fails).
  2. \( 4(5) + 4(15) \leq 100 \) (Passes).

Correct answer: No, because the bike order does not meet the restrictions of 4c + 6a ≤ 120 and 4c + 4a ≤ 100.

Question 9

The shading areas must correspond to:

  • A solid line \( g \) representing an inequality below.
  • A dashed line \( f \) shading above.

From given points and shading, assess valid points (like \( y > -x + 3 \) etc).

Correct answer: y > −x + 3

Question 10

To establish inequalities:

  1. Room for people: \( x + y ≤ 200 \)
  2. Budget constraint: \( 4x + 3y ≤ 750 \)

Correct answer: x + y ≤ 200; 4x + 3y ≤ 750

If you have further details you'd like to explore or discuss, please let me know!