Question
A teacher asks a student to write and solve an inequality to find what numbers satisfy the statement that 3.2 more than 7 times a number, n is less than 17.2. The table shows the student’s work.
step 1. 7n + 3.2 < 17.2
step 2. 7n < 14
step 3. n < 7
Which statement is true?
The student’s work in all of the steps is correct because all numbers less than 7 satisfy the statement.
The student made a mistake writing the inequality before Step 1. The correct inequality that represents the statement is 3.2n+7<17.2
The student made a mistake between Step 1. and Step 2. by subtracting 3.2 from both sides of the inequality instead of adding 3.2 to both sides of the inequality.
The student made a mistake between Step 2. and Step 3. by subtracting 7 from the right side of the inequality instead of dividing both sides of the inequality by 7.
step 1. 7n + 3.2 < 17.2
step 2. 7n < 14
step 3. n < 7
Which statement is true?
The student’s work in all of the steps is correct because all numbers less than 7 satisfy the statement.
The student made a mistake writing the inequality before Step 1. The correct inequality that represents the statement is 3.2n+7<17.2
The student made a mistake between Step 1. and Step 2. by subtracting 3.2 from both sides of the inequality instead of adding 3.2 to both sides of the inequality.
The student made a mistake between Step 2. and Step 3. by subtracting 7 from the right side of the inequality instead of dividing both sides of the inequality by 7.
Answers
GPT-4o mini
Let's analyze the student's steps to determine the correct answer.
The original statement is: "3.2 more than 7 times a number, n, is less than 17.2."
This can be translated into an inequality:
1. **Step 1 is correct:** The student correctly wrote the inequality as \( 7n + 3.2 < 17.2 \).
2. **Step 2:** To isolate \( n \), the student needs to subtract 3.2 from both sides.
- Starting from \( 7n + 3.2 < 17.2 \), if we subtract 3.2 from both sides, we get:
\[
7n < 17.2 - 3.2
\]
\[
7n < 14
\]
This step is also correct.
3. **Step 3:** Now to solve for \( n \), the student should divide both sides of the inequality \( 7n < 14 \) by 7.
- This gives:
\[
n < \frac{14}{7}
\]
\[
n < 2
\]
The student mistakenly wrote \( n < 7 \) instead.
Now, let's analyze the truth of the provided statements:
1. **The student’s work in all of the steps is correct because all numbers less than 7 satisfy the statement.**
- This is FALSE because \( n < 2 \) is the correct solution.
2. **The student made a mistake writing the inequality before Step 1. The correct inequality that represents the statement is \( 3.2n + 7 < 17.2 \)**
- This is FALSE because the original inequality written in Step 1 is correct.
3. **The student made a mistake between Step 1 and Step 2 by subtracting 3.2 from both sides of the inequality instead of adding 3.2 to both sides of the inequality.**
- This is FALSE because the student correctly subtracted 3.2.
4. **The student made a mistake between Step 2 and Step 3 by subtracting 7 from the right side of the inequality instead of dividing both sides of the inequality by 7.**
- This is TRUE. The student incorrectly jumped to \( n < 7 \) instead of correctly dividing by 7.
Therefore, the correct statement is: **The student made a mistake between Step 2 and Step 3 by subtracting 7 from the right side of the inequality instead of dividing both sides of the inequality by 7.**
The original statement is: "3.2 more than 7 times a number, n, is less than 17.2."
This can be translated into an inequality:
1. **Step 1 is correct:** The student correctly wrote the inequality as \( 7n + 3.2 < 17.2 \).
2. **Step 2:** To isolate \( n \), the student needs to subtract 3.2 from both sides.
- Starting from \( 7n + 3.2 < 17.2 \), if we subtract 3.2 from both sides, we get:
\[
7n < 17.2 - 3.2
\]
\[
7n < 14
\]
This step is also correct.
3. **Step 3:** Now to solve for \( n \), the student should divide both sides of the inequality \( 7n < 14 \) by 7.
- This gives:
\[
n < \frac{14}{7}
\]
\[
n < 2
\]
The student mistakenly wrote \( n < 7 \) instead.
Now, let's analyze the truth of the provided statements:
1. **The student’s work in all of the steps is correct because all numbers less than 7 satisfy the statement.**
- This is FALSE because \( n < 2 \) is the correct solution.
2. **The student made a mistake writing the inequality before Step 1. The correct inequality that represents the statement is \( 3.2n + 7 < 17.2 \)**
- This is FALSE because the original inequality written in Step 1 is correct.
3. **The student made a mistake between Step 1 and Step 2 by subtracting 3.2 from both sides of the inequality instead of adding 3.2 to both sides of the inequality.**
- This is FALSE because the student correctly subtracted 3.2.
4. **The student made a mistake between Step 2 and Step 3 by subtracting 7 from the right side of the inequality instead of dividing both sides of the inequality by 7.**
- This is TRUE. The student incorrectly jumped to \( n < 7 \) instead of correctly dividing by 7.
Therefore, the correct statement is: **The student made a mistake between Step 2 and Step 3 by subtracting 7 from the right side of the inequality instead of dividing both sides of the inequality by 7.**