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 made a mistake going from step 1 to step 2


Izzie made no mistakes in solving this inequality


The correct answer should have been t≤8


Izzie made a mistake going from step 3 to step 4


Izzie made a mistake going from step 2 to step 3


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

Answer 1

Let's go through Izzie's steps to verify whether she solved the inequality correctly:

The original inequality is:
\[ 3(t + 1) - 4t \geq -5 \]

**Step 1: Expand and simplify**
\[ 3t + 3 - 4t \geq -5 \]
This simplifies to:
\[ -t + 3 \geq -5 \]
(So Step 1 to Step 2 is correct.)

**Step 2: Rearranging to isolate \( t \)**
\[ -t + 3 \geq -5 \]
We subtract 3 from both sides:
\[ -t \geq -8 \]

**Step 3: Dividing by -1 (this reverses the inequality)**
When you divide or multiply an inequality by a negative number, you must reverse the inequality symbol:
\[ t \leq 8 \]
(So Step 3 to Step 4 is not correct, as Izzie didn't change the inequality direction when multiplying by -1.)

Now, let's evaluate the statements:

1. **Izzie made a mistake going from step 1 to step 2:**
- **False**; she correctly simplified in step 1.

2. **Izzie made no mistakes in solving this inequality:**
- **False**; she made a mistake in step 3.

3. **The correct answer should have been \( t \leq 8 \):**
- **True**; this is the correct solution.

4. **Izzie made a mistake going from step 3 to step 4:**
- **True**; she didn’t correctly reverse the inequality.

5. **Izzie made a mistake going from step 2 to step 3:**
- **False**; the transition was correct, but she mismanaged the inequality in the next step.

6. **Izzie should not have changed the inequality symbol from \( \geq \) to \( \leq \):**
- **True**; she should have reversed the direction of the inequality when dividing by -1.

So, the final answers to the statements are:

- Izzie made a mistake going from step 1 to step 2: **False**
- Izzie made no mistakes in solving this inequality: **False**
- The correct answer should have been \( t \leq 8 \): **True**
- Izzie made a mistake going from step 3 to step 4: **True**
- Izzie made a mistake going from step 2 to step 3: **False**
- Izzie should not have changed the inequality symbol from \( \geq \) to \( \leq \): **True**