Question

Show which of the following points is part of the solution set for the inequality y≤25x+1 .(1 point)
Responses

(10,4)
left parenthesis 10 comma 4 right parenthesis

(−10,6)
left parenthesis negative 10 comma 6 right parenthesis

(10,6)
left parenthesis 10 comma 6 right parenthesis

(10,5.5)

Answer 1

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

(2,4)
left parenthesis 2 comma 4 right parenthesis

(6,6)
left parenthesis 6 comma 6 right parenthesis

(1,4.5)
left parenthesis 1 comma 4.5 right parenthesis

(−2,6)

Answer 2

Use the image to answer the question.

The x- and y- axes of a coordinate plane range from 0 to 30 and 0 to 20 in increments of 5 and are labeled 'Lawns' and 'Dogs,' respectively. A line connects left parenthesis 0 comma 10 right parenthesis and left parenthesis 8 comma 0 right parenthesis.

This graph shows the inequality 25x+15y≥200, and represents 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

(2,3)
left parenthesis 2 comma 3 right parenthesis

(10,10)
left parenthesis 10 comma 10 right parenthesis

(20,11.5)
left parenthesis 20 comma 11.5 right parenthesis

None of the points are a viable solution.

Answer 3

To determine which of the given points is part of the solution set for the inequality \( y \leq 25x + 1 \), we need to substitute each point into the inequality and see if the inequality holds true.

1. **Point (10, 4)**:
Substitute \( x = 10 \) into the equation \( y = 25(10) + 1 = 250 + 1 = 251 \).
Check: \( 4 \leq 251 \) (True)

2. **Point (−10, 6)**:
Substitute \( x = -10 \) into the equation \( y = 25(-10) + 1 = -250 + 1 = -249 \).
Check: \( 6 \leq -249 \) (False)

3. **Point (10, 6)**:
Substitute \( x = 10 \) into the equation \( y = 25(10) + 1 = 251 \).
Check: \( 6 \leq 251 \) (True)

4. **Point (10, 5.5)**:
Substitute \( x = 10 \) into the equation \( y = 25(10) + 1 = 251 \).
Check: \( 5.5 \leq 251 \) (True)

Now we note the results:
- (10, 4) is valid.
- (−10, 6) is invalid.
- (10, 6) is valid.
- (10, 5.5) is valid.

Therefore, the points (10, 4), (10, 6), and (10, 5.5) are all part of the solution set for the inequality \( y \leq 25x + 1 \).

If you need just one point, you can say:

**Valid Points: (10, 4), (10, 6), (10, 5.5)**.

Answer 4

Let's define the variables involved in the problem:

- Let \( x \) be the number of hardbacks that Kiani buys.
- Let \( y \) be the number of paperbacks that Kiani buys.

### Writing a Two-Variable Inequality

Kiani's budget constraint can be expressed as follows:

\[ 12x + 5y \leq 60 \]

This inequality states that the total cost of hardbacks and paperbacks must be less than or equal to Kiani's budget of $60.

### Constraints of the Problem

The constraints for the problem also involve the non-negativity of the quantities purchased, so we have:

1. \( x \geq 0 \) (Kiani cannot buy a negative number of hardbacks)
2. \( y \geq 0 \) (Kiani cannot buy a negative number of paperbacks)

Thus, the complete set of constraints is:

1. \( 12x + 5y \leq 60 \)
2. \( x \geq 0 \)
3. \( y \geq 0 \)

### Analyzing the Given Points

Now let's evaluate each of the points provided in the options to see if they satisfy the inequality \( 12x + 5y \leq 60 \):

1. **Point (2, 4)**:
\( 12(2) + 5(4) = 24 + 20 = 44 \)
Check: \( 44 \leq 60 \) (True)

2. **Point (6, 6)**:
\( 12(6) + 5(6) = 72 + 30 = 102 \)
Check: \( 102 \leq 60 \) (False)

3. **Point (1, 4.5)**:
\( 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \)
Check: \( 34.5 \leq 60 \) (True)

4. **Point (−2, 6)**:
\( x \) is negative, so this point is not viable regardless of the cost.
Check: Invalid

### Conclusion

The viable solutions that satisfy the constraints are:

- **Point (2, 4) is valid.**
- **Point (1, 4.5) is valid.**

However, the only point that matches the requirement for a viable solution from the options provided is:

**(2, 4)**.

Answer 5

To determine which of the provided points is a viable solution to the inequality \( 25x + 15y \geq 200 \) in the context of Lina's earnings for lawn mowing and dog walking, we'll evaluate each point by substituting the values of \( x \) and \( y \) into the inequality.

### Evaluating Each Point

1. **Point (2, 3)**:
\[
25(2) + 15(3) = 50 + 45 = 95
\]
Check: \( 95 \geq 200 \) (False)

2. **Point (10, 10)**:
\[
25(10) + 15(10) = 250 + 150 = 400
\]
Check: \( 400 \geq 200 \) (True)

3. **Point (20, 11.5)**:
\[
25(20) + 15(11.5) = 500 + 172.5 = 672.5
\]
Check: \( 672.5 \geq 200 \) (True)

### Conclusion

Among the points evaluated:

- **Point (2, 3)** is not a viable solution.
- **Point (10, 10)** is a viable solution.
- **Point (20, 11.5)** is also a viable solution.

Since both (10, 10) and (20, 11.5) are viable solutions, if you are restricted to only picking one option, **(10, 10)** is a valid response since the prompt asks for a single viable solution.

So, the final response would be **(10, 10)**.