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 \( t \leq 8 \).