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


The correct answer should have been t≤8


Izzie made no mistakes in solving this inequality


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


Izzie made a mistake going from step 3 to step 4


Izzie made a mistake going from step 2 to step 3

Answer 1

Let's analyze Izzie's work step by step:

1. **Step 1**: \( 3(t + 1) - 4t \geq -5 \) simplifies to \( 3t + 3 - 4t \geq -5 \). This is correct and simplifies to \( -t + 3 \geq -5 \).

2. **Step 2**: This matches Izzie's completion: \( -t + 3 \geq -5 \). This step is also correct.

3. **Step 3**: Izzie writes \( -t \geq -2 \). To isolate \( t \), you multiply or divide by -1, which reverses the inequality. So this step should become \( t \leq 2 \). This step is correct.

4. **Step 4**: She states \( t \leq 2 \). This is indeed the correct representation of the solution after considering the flipped inequality from Step 3.

Now let's evaluate the statements:

1. **Izzie made a mistake going from step 1 to step 2**: **False**. She did not make a mistake.

2. **The correct answer should have been \( t \leq 8 \)**: **False**. The correct answer is \( t \leq 2 \).

3. **Izzie made no mistakes in solving this inequality**: **True**. She solved it correctly.

4. **Izzie should not have changed the inequality symbol from \( \geq \) to \( \leq \)**: **False**. She appropriately changed the inequality when multiplying by -1.

5. **Izzie made a mistake going from step 3 to step 4**: **False**. There was no mistake in the transition from Step 3 to Step 4.

6. **Izzie made a mistake going from step 2 to step 3**: **False**. She did not make a mistake there.

Overall, Izzie's work is correct.