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 or False
Izzie made no mistakes in solving this inequality


Izzie made a mistake going from step 1 to step 2


Izzie made a mistake going from step 2 to step 3


The correct answer should have been t≤8


Izzie should not have changed the inequality symbol from ≥
to ≤


Izzie made a mistake going from step 3 to step 4

Answer 1

Let’s go through Izzie’s work step by step to verify the correctness of her solution.

Starting with the original inequality:
\[3(t+1)−4t≥−5\]

**Step 1:**
Distributing \(3\) in the expression gives:
\[3t + 3 - 4t ≥ -5\]
This step is correct.

**Step 2:**
Combining like terms results in:
\[-t + 3 ≥ -5\]
This step is correct as well.

**Step 3:**
To isolate \(-t\), we subtract \(3\) from both sides:
\[-t ≥ -5 - 3\]
\[-t ≥ -8\]

Now, to resolve \(-t ≥ -8\), we must divide by \(-1\). Remember that when we divide or multiply by a negative number, the inequality sign changes direction:
\[t ≤ 8\]
So, this is where Izzie went wrong; she did not change the inequality sign.

**Step 4:**
Izzie's conclusion of \(t ≤ 2\) is incorrect.

Now, summarizing the evaluations of her statements:

1. **Izzie made no mistakes in solving this inequality**: False (she made a mistake in changing the inequality sign).

2. **Izzie made a mistake going from step 1 to step 2**: False (step 1 to step 2 is correct).

3. **Izzie made a mistake going from step 2 to step 3**: True (she failed to change the sign correctly after multiplying by \(-1\)).

4. **The correct answer should have been t ≤ 8**: True (that is the correct answer).

5. **Izzie should not have changed the inequality symbol from ≥ to ≤**: True (she made a mistake in doing so).

6. **Izzie made a mistake going from step 3 to step 4**: True (she incorrectly derived \(t ≤ 2\) instead of \(t ≤ 8\)).

To recap:

- Overall, Izzie made mistakes in the steps that affected her conclusion.
- The incorrect interpretation of the inequality was the primary error.

So the conclusions above about her steps and correctness are valid based on the analysis we performed.