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 ≤


Izzie made a mistake going from step 3 to step 4


The correct answer should have been t≤8


Izzie made no mistakes in solving this inequality


Izzie made a mistake going from step 2 to step 3


Izzie made a mistake going from step 1 to step 2

Answer 1

Let's evaluate Izzie's work step by step to determine whether she correctly solved the inequality:

1. **Original Inequality**:
\( 3(t+1) - 4t \geq -5 \)

2. **Step 1**:
Distributing the \(3\):
\[
3t + 3 - 4t \geq -5
\]
This step is correct.

3. **Step 2**:
Combine like terms:
\[
-t + 3 \geq -5
\]
This step is also correct.

4. **Step 3**:
To isolate \(t\), subtract \(3\) from both sides:
\[
-t \geq -2
\]
This step is correct too.

5. **Step 4**:
Now, to solve for \(t\), multiply both sides by \(-1\). When you do this, you must reverse the inequality symbol:
\[
t \leq 2
\]
This step is where she made a mistake because she failed to reverse the inequality symbol when multiplying by \(-1\).

Now, let's address the statements:

1. **True/False**: Izzie should not have changed the inequality symbol from ≥ to ≤.
- **True**. She should have reversed the inequality symbol.

2. **True/False**: Izzie made a mistake going from step 3 to step 4.
- **True**. She did make a mistake in applying the inequality rule.

3. **True/False**: The correct answer should have been t≤8.
- **False**. The correct answer should have been \(t \geq 2\).

4. **True/False**: Izzie made no mistakes in solving this inequality.
- **False**. She made mistakes.

5. **True/False**: Izzie made a mistake going from step 2 to step 3.
- **False**. There was no mistake from step 2 to step 3.

6. **True/False**: Izzie made a mistake going from step 1 to step 2.
- **False**. There was no mistake from step 1 to step 2.

To summarize:
- The statements are:
1. True
2. True
3. False
4. False
5. False
6. False