Question
Justin has $110 to spend at the mall. He wants to buy a hat that costs $25 and a pair of jeans that cost $40. He is also thinking about buying some bags of socks that cost $12 each. There is no tax on any of his purchases.
Which statements are true?
Choose all that apply.
If x is the number of bags of socks Justin can buy, then the inequality representing this situation is x < 3.
If Justin buys the hat and the jeans, solving the inequality
for x will tell him the maximum number of bags of socks he can buy.
If Justin buys the hat but not the jeans, solving the inequality
for x will tell him the maximum number of bags of socks he can buy.
If Justin buys the hats, jeans, and the maximum number of bags of socks, he will have enough money left to buy a belt that is on sale for $7.99.
A.
If x is the number of bags of socks Justin can buy, then the inequality representing this situation is x < 3.
B.
If Justin buys the hat and the jeans, solving the inequality
for x will tell him the maximum number of bags of socks he can buy.
C.
If Justin buys the hat but not the jeans, solving the inequality
for x will tell him the maximum number of bags of socks he can buy.
D.
If Justin buys the hats, jeans, and the maximum number of bags of socks, he will have enough money left to buy a belt that is on sale for $7.99.
Which statements are true?
Choose all that apply.
If x is the number of bags of socks Justin can buy, then the inequality representing this situation is x < 3.
If Justin buys the hat and the jeans, solving the inequality
for x will tell him the maximum number of bags of socks he can buy.
If Justin buys the hat but not the jeans, solving the inequality
for x will tell him the maximum number of bags of socks he can buy.
If Justin buys the hats, jeans, and the maximum number of bags of socks, he will have enough money left to buy a belt that is on sale for $7.99.
A.
If x is the number of bags of socks Justin can buy, then the inequality representing this situation is x < 3.
B.
If Justin buys the hat and the jeans, solving the inequality
for x will tell him the maximum number of bags of socks he can buy.
C.
If Justin buys the hat but not the jeans, solving the inequality
for x will tell him the maximum number of bags of socks he can buy.
D.
If Justin buys the hats, jeans, and the maximum number of bags of socks, he will have enough money left to buy a belt that is on sale for $7.99.
Answers
GPT-4o mini
To analyze the situation, let's break down Justin's budget and expenses.
1. Justin has $110 to spend.
2. The hat costs $25.
3. The jeans cost $40.
4. Each bag of socks costs $12.
### Determining the statements:
1. **Statement A: If x is the number of bags of socks Justin can buy, then the inequality representing this situation is x < 3.**
Let's find the total cost if Justin buys the hat and jeans:
\[
\text{Cost of hat and jeans} = 25 + 40 = 65
\]
The remaining money after buying the hat and jeans:
\[
\text{Remaining money} = 110 - 65 = 45
\]
The cost of the socks:
\[
12x \leq 45
\]
Solving this gives:
\[
x \leq \frac{45}{12} = 3.75
\]
Thus, the maximum whole number of bags Justin can buy is 3, and the inequality is actually \(x \leq 3\) and not \(x < 3\).
**Statement A is FALSE.**
2. **Statement B: If Justin buys the hat and the jeans, solving the inequality for x will tell him the maximum number of bags of socks he can buy.**
We have just established that if he buys the hat and jeans, he has $45 left to spend on socks, and solving \(12x \leq 45\) will indeed give the maximum number of bags of socks he can buy.
**Statement B is TRUE.**
3. **Statement C: If Justin buys the hat but not the jeans, solving the inequality for x will tell him the maximum number of bags of socks he can buy.**
If he buys only the hat:
\[
\text{Remaining money} = 110 - 25 = 85
\]
The inequality for socks becomes:
\[
12x \leq 85
\]
This will help him find out how many bags of socks he can buy in this case as well.
**Statement C is TRUE.**
4. **Statement D: If Justin buys the hat, jeans, and the maximum number of bags of socks, he will have enough money left to buy a belt that is on sale for $7.99.**
If Justin buys the maximum of 3 bags of socks after buying the hat and jeans:
\[
\text{Total spent on socks} = 12 \times 3 = 36
\]
Total spent overall:
\[
\text{Total spent} = 25 + 40 + 36 = 101
\]
Remaining money:
\[
110 - 101 = 9
\]
Since the belt costs $7.99, Justin can afford it with the remaining $9.
**Statement D is TRUE.**
### Summary of the true statements:
- B: TRUE
- C: TRUE
- D: TRUE
**Final answers: B, C, and D are correct.**
1. Justin has $110 to spend.
2. The hat costs $25.
3. The jeans cost $40.
4. Each bag of socks costs $12.
### Determining the statements:
1. **Statement A: If x is the number of bags of socks Justin can buy, then the inequality representing this situation is x < 3.**
Let's find the total cost if Justin buys the hat and jeans:
\[
\text{Cost of hat and jeans} = 25 + 40 = 65
\]
The remaining money after buying the hat and jeans:
\[
\text{Remaining money} = 110 - 65 = 45
\]
The cost of the socks:
\[
12x \leq 45
\]
Solving this gives:
\[
x \leq \frac{45}{12} = 3.75
\]
Thus, the maximum whole number of bags Justin can buy is 3, and the inequality is actually \(x \leq 3\) and not \(x < 3\).
**Statement A is FALSE.**
2. **Statement B: If Justin buys the hat and the jeans, solving the inequality for x will tell him the maximum number of bags of socks he can buy.**
We have just established that if he buys the hat and jeans, he has $45 left to spend on socks, and solving \(12x \leq 45\) will indeed give the maximum number of bags of socks he can buy.
**Statement B is TRUE.**
3. **Statement C: If Justin buys the hat but not the jeans, solving the inequality for x will tell him the maximum number of bags of socks he can buy.**
If he buys only the hat:
\[
\text{Remaining money} = 110 - 25 = 85
\]
The inequality for socks becomes:
\[
12x \leq 85
\]
This will help him find out how many bags of socks he can buy in this case as well.
**Statement C is TRUE.**
4. **Statement D: If Justin buys the hat, jeans, and the maximum number of bags of socks, he will have enough money left to buy a belt that is on sale for $7.99.**
If Justin buys the maximum of 3 bags of socks after buying the hat and jeans:
\[
\text{Total spent on socks} = 12 \times 3 = 36
\]
Total spent overall:
\[
\text{Total spent} = 25 + 40 + 36 = 101
\]
Remaining money:
\[
110 - 101 = 9
\]
Since the belt costs $7.99, Justin can afford it with the remaining $9.
**Statement D is TRUE.**
### Summary of the true statements:
- B: TRUE
- C: TRUE
- D: TRUE
**Final answers: B, C, and D are correct.**