Question
Izzie is trying to solve this inequality: 3(t+1)−4t≥−5
After she solves it, she shows it to you and asks you if she did it correctly.
This is the work she completed:
Step 1: 3t+3−4t≥−5
Step 2: −t+3≥−5
Step 3: −t≥−2
Step 4: t≤2
(12 points)
True False
Izzie should not have changed the inequality symbol from ≥
to ≤
The correct answer should have been t≤8
Izzie made a mistake going from step 2 to step 3
Izzie made a mistake going from step 3 to step 4
Izzie made a mistake going from step 1 to step 2
Izzie made no mistakes in solving this inequality
After she solves it, she shows it to you and asks you if she did it correctly.
This is the work she completed:
Step 1: 3t+3−4t≥−5
Step 2: −t+3≥−5
Step 3: −t≥−2
Step 4: t≤2
(12 points)
True False
Izzie should not have changed the inequality symbol from ≥
to ≤
The correct answer should have been t≤8
Izzie made a mistake going from step 2 to step 3
Izzie made a mistake going from step 3 to step 4
Izzie made a mistake going from step 1 to step 2
Izzie made no mistakes in solving this inequality
Answers
Answered by
GPT-4o mini
Let's go through Izzie's steps to verify whether she solved the inequality correctly.
Starting with the original inequality:
\[ 3(t+1) - 4t \geq -5 \]
**Step 1: Distributing the 3**
\[ 3(t + 1) - 4t \geq -5 \]
\[ 3t + 3 - 4t \geq -5 \]
This step is correct.
**Step 2: Combining like terms**
\[ 3t + 3 - 4t \geq -5 \]
Combining \(3t\) and \(-4t\):
\[ -t + 3 \geq -5 \]
This step is correct.
**Step 3: Isolating the variable \(-t\)**
Subtracting 3 from both sides:
\[ -t \geq -5 - 3 \]
Which simplifies to:
\[ -t \geq -8 \]
This is where Izzie made a mistake as she stated:
\[ -t \geq -2 \]
**Step 4: Determining \(t\)**
If we negate both sides of the correct result \( -t \geq -8 \), we must flip the inequality sign:
\[ t \leq 8 \]
In Izzie's Step 4, she concluded that \(t \leq 2\), which is incorrect based on the previous miscalculation.
---
### Summary of Evaluations:
1. **Izzie should not have changed the inequality symbol from ≥ to ≤**: **True** (The sign flips when multiplying or dividing by a negative number.)
2. **The correct answer should have been t ≤ 8**: **True**
3. **Izzie made a mistake going from Step 2 to Step 3**: **True** (Her calculation was incorrect; she miscalculated.)
4. **Izzie made a mistake going from Step 3 to Step 4**: **True** (Her conclusion that \(t \leq 2\) was based on an earlier error.)
5. **Izzie made a mistake going from Step 1 to Step 2**: **False** (She correctly simplified the expression.)
6. **Izzie made no mistakes in solving this inequality**: **False** (She made a mistake in her calculations.)
So, the correct answers to each statement about Izzie’s work are True, True, True, True, False, False.
Starting with the original inequality:
\[ 3(t+1) - 4t \geq -5 \]
**Step 1: Distributing the 3**
\[ 3(t + 1) - 4t \geq -5 \]
\[ 3t + 3 - 4t \geq -5 \]
This step is correct.
**Step 2: Combining like terms**
\[ 3t + 3 - 4t \geq -5 \]
Combining \(3t\) and \(-4t\):
\[ -t + 3 \geq -5 \]
This step is correct.
**Step 3: Isolating the variable \(-t\)**
Subtracting 3 from both sides:
\[ -t \geq -5 - 3 \]
Which simplifies to:
\[ -t \geq -8 \]
This is where Izzie made a mistake as she stated:
\[ -t \geq -2 \]
**Step 4: Determining \(t\)**
If we negate both sides of the correct result \( -t \geq -8 \), we must flip the inequality sign:
\[ t \leq 8 \]
In Izzie's Step 4, she concluded that \(t \leq 2\), which is incorrect based on the previous miscalculation.
---
### Summary of Evaluations:
1. **Izzie should not have changed the inequality symbol from ≥ to ≤**: **True** (The sign flips when multiplying or dividing by a negative number.)
2. **The correct answer should have been t ≤ 8**: **True**
3. **Izzie made a mistake going from Step 2 to Step 3**: **True** (Her calculation was incorrect; she miscalculated.)
4. **Izzie made a mistake going from Step 3 to Step 4**: **True** (Her conclusion that \(t \leq 2\) was based on an earlier error.)
5. **Izzie made a mistake going from Step 1 to Step 2**: **False** (She correctly simplified the expression.)
6. **Izzie made no mistakes in solving this inequality**: **False** (She made a mistake in her calculations.)
So, the correct answers to each statement about Izzie’s work are True, True, True, True, False, False.