Question

14. 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 2 to step 3


Izzie made a mistake going from step 3 to step 4


The correct answer should have been t≤8


Izzie made a mistake going from step 1 to step 2


Izzie made no mistakes in solving this inequality


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

Answer 1

Let's go through Izzie's work step-by-step to determine if she made any mistakes.

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

### Step 1: Distribute and combine like terms
Izzie rewrote this as:
\[ 3t + 3 - 4t \geq -5 \]
which simplifies to:
\[ -t + 3 \geq -5 \]

This step is **correct**.

### Step 2: Move constant terms to the right
From:
\[ -t + 3 \geq -5 \]
Izzie went to:
\[ -t \geq -5 - 3 \]
or:
\[ -t \geq -8 \]
but she wrote:
\[ -t + 3 \geq -5 \]
to:
\[ -t \geq -2 \]

**This step contains a mistake.** She should have subtracted 3 from both sides, leading to:
\[ -t \geq -8 \]

### Step 3: Flip the inequality sign when multiplying/dividing
Izzie started with:
\[ -t \geq -2 \]
and moved to:
\[ t \leq 2 \]
This is incorrect because, starting from the corrected inequality \( -t \geq -8 \), when dividing by -1, the inequality sign flips:
\[ t \leq 8 \]
Thus, she made a mistake here.

### Summary of responses:
- **Izzie made a mistake going from step 2 to step 3**: **True**
- **Izzie made a mistake going from step 3 to step 4**: **True** (as it's based on an earlier mistake)
- **The correct answer should have been \( t \leq 8 \)**: **True**
- **Izzie made a mistake going from step 1 to step 2**: **False**
- **Izzie made no mistakes in solving this inequality**: **False**
- **Izzie should not have changed the inequality symbol from \( \geq \) to \( \leq \)**: **True** (but this is a consequence of making a mistake when solving)

In conclusion, Izzie made mistakes in her solution, particularly in Step 2 and Step 3, and the correct answer should be \( t \leq 8 \).