Question
Question 1 A)Which of the following points is a solution to the linear inequality y>19x−3y>19x−3(1 point) Responses (0,-1) (0,-1) (0,-3) (0,-3) (0,-4) (0,-4) (0,-6) (0,-6) Question 2 A) Use the image to answer the question. Is the point (0,-3) a solution to the inequality y>45x−3y>45x−3, as shown on the graph? (1 point) Responses Yes, because (0,-3) falls on the boundary line Yes, because (0,-3) falls on the boundary line No, because (0,-3) falls on the boundary line No, because (0,-3) falls on the boundary line Yes, because (0,-3) falls within the shaded region Yes, because (0,-3) falls within the shaded region No, because (0,-3) falls within the shaded region No, because (0,-3) falls within the shaded region Question 3 A)Show which of the following points is part of the solution set for the inequality y≤25x+1y≤25x+1(1 point) Responses (10,6) (10,6) (-10,6) (-10,6) (10,5.5) (10,5.5) (10,4) (10,4) Question 4 A)Kiani is shopping at a bookstore with a budget of $60. Hardbacks cost $12 and paperbacks cost $5. Write a two-variable inequality and determine the constraints of the problem. Which of the following points represents a viable solution?(1 point) Responses (6,6) (6,6) (2,4) (2,4) (-2,6) (-2,6) (1,4.5) (1,4.5) Question 5 A) Use the image to answer the question. This graph shows the inequality 25x+15y≥20025x+15y≥200, representing Lina's earnings for lawn mowing and dog walking, where x represents the number of lawns mowed and y represents the number of dogs walked. Which of the labeled points is a viable solution in this context? (1 point) Responses None of the points area viable solution None of the points area viable solution (2,3) (2,3) (10,10) (10,10) (20,11.5) (20,11.5)
Answers
GPT-4o mini
Let's go through each question one by one:
### Question 1 A)
**Inequality:** \( y > 19x - 3 \)
To determine if a point is a solution, we'll substitute \( x = 0 \) into the inequality and check the resulting value of \( y \):
- For (0, -1):
\( -1 > 19(0) - 3 \) → \( -1 > -3 \) (True)
- For (0, -3):
\( -3 > 19(0) - 3 \) → \( -3 > -3 \) (False)
- For (0, -4):
\( -4 > 19(0) - 3 \) → \( -4 > -3 \) (False)
- For (0, -6):
\( -6 > 19(0) - 3 \) → \( -6 > -3 \) (False)
**Answer: (0, -1)**
---
### Question 2 A)
**Inequality:** \( y > 45x - 3 \)
Given the graph, we will determine whether the point (0, -3) is a solution.
- Since (0, -3) is on the boundary line, it cannot be a solution to \( y > 45x - 3 \). The inequality is strict ('greater than'), so points on the line itself do not satisfy the inequality.
**Answer: No, because (0,-3) falls on the boundary line**
---
### Question 3 A)
**Inequality:** \( y \leq 25x + 1 \)
We will check each point to see if it satisfies the inequality:
- For (10, 6):
\( 6 \leq 25(10) + 1 \) → \( 6 \leq 251 \) (True)
- For (-10, 6):
\( 6 \leq 25(-10) + 1 \) → \( 6 \leq -249 \) (False)
- For (10, 5.5):
\( 5.5 \leq 25(10) + 1 \) → \( 5.5 \leq 251 \) (True)
- For (10, 4):
\( 4 \leq 25(10) + 1 \) → \( 4 \leq 251 \) (True)
The point (10, 6) is part of the solution set along with (10, 5.5) and (10, 4).
**Answer: (10, 6)**
---
### Question 4 A)
**Constraints:** Kiani has $60 to spend on hardbacks ($12 each) and paperbacks ($5 each). Let \( x \) be the number of hardbacks and \( y \) the number of paperbacks.
The inequality can be formulated as:
\( 12x + 5y \leq 60 \)
Now we can evaluate the response options considering they need to be non-negative integers:
- For (6, 6):
\( 12(6) + 5(6) = 72 + 30 = 102 \) (False)
- For (2, 4):
\( 12(2) + 5(4) = 24 + 20 = 44 \) (True)
- For (-2, 6):
Not viable (negative quantity).
- For (1, 4.5):
Not viable (non-integer).
**Answer: (2, 4)**
---
### Question 5 A)
**Inequality:** \( 25x + 15y \geq 200 \)
We need to evaluate which of the points meet this condition:
- For (2, 3):
\( 25(2) + 15(3) = 50 + 45 = 95 \) (False)
- For (10, 10):
\( 25(10) + 15(10) = 250 + 150 = 400 \) (True)
- For (20, 11.5):
\( 25(20) + 15(11.5) = 500 + 172.5 = 672.5 \) (True)
Since (10, 10) and (20, 11.5) satisfy the inequality.
**Answer: (10, 10) is a viable solution.**
### Question 1 A)
**Inequality:** \( y > 19x - 3 \)
To determine if a point is a solution, we'll substitute \( x = 0 \) into the inequality and check the resulting value of \( y \):
- For (0, -1):
\( -1 > 19(0) - 3 \) → \( -1 > -3 \) (True)
- For (0, -3):
\( -3 > 19(0) - 3 \) → \( -3 > -3 \) (False)
- For (0, -4):
\( -4 > 19(0) - 3 \) → \( -4 > -3 \) (False)
- For (0, -6):
\( -6 > 19(0) - 3 \) → \( -6 > -3 \) (False)
**Answer: (0, -1)**
---
### Question 2 A)
**Inequality:** \( y > 45x - 3 \)
Given the graph, we will determine whether the point (0, -3) is a solution.
- Since (0, -3) is on the boundary line, it cannot be a solution to \( y > 45x - 3 \). The inequality is strict ('greater than'), so points on the line itself do not satisfy the inequality.
**Answer: No, because (0,-3) falls on the boundary line**
---
### Question 3 A)
**Inequality:** \( y \leq 25x + 1 \)
We will check each point to see if it satisfies the inequality:
- For (10, 6):
\( 6 \leq 25(10) + 1 \) → \( 6 \leq 251 \) (True)
- For (-10, 6):
\( 6 \leq 25(-10) + 1 \) → \( 6 \leq -249 \) (False)
- For (10, 5.5):
\( 5.5 \leq 25(10) + 1 \) → \( 5.5 \leq 251 \) (True)
- For (10, 4):
\( 4 \leq 25(10) + 1 \) → \( 4 \leq 251 \) (True)
The point (10, 6) is part of the solution set along with (10, 5.5) and (10, 4).
**Answer: (10, 6)**
---
### Question 4 A)
**Constraints:** Kiani has $60 to spend on hardbacks ($12 each) and paperbacks ($5 each). Let \( x \) be the number of hardbacks and \( y \) the number of paperbacks.
The inequality can be formulated as:
\( 12x + 5y \leq 60 \)
Now we can evaluate the response options considering they need to be non-negative integers:
- For (6, 6):
\( 12(6) + 5(6) = 72 + 30 = 102 \) (False)
- For (2, 4):
\( 12(2) + 5(4) = 24 + 20 = 44 \) (True)
- For (-2, 6):
Not viable (negative quantity).
- For (1, 4.5):
Not viable (non-integer).
**Answer: (2, 4)**
---
### Question 5 A)
**Inequality:** \( 25x + 15y \geq 200 \)
We need to evaluate which of the points meet this condition:
- For (2, 3):
\( 25(2) + 15(3) = 50 + 45 = 95 \) (False)
- For (10, 10):
\( 25(10) + 15(10) = 250 + 150 = 400 \) (True)
- For (20, 11.5):
\( 25(20) + 15(11.5) = 500 + 172.5 = 672.5 \) (True)
Since (10, 10) and (20, 11.5) satisfy the inequality.
**Answer: (10, 10) is a viable solution.**