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:
-
Izzie made a mistake going from step 1 to step 2:
- False; she correctly simplified in step 1.
-
Izzie made no mistakes in solving this inequality:
- False; she made a mistake in step 3.
-
The correct answer should have been \( t \leq 8 \):
- True; this is the correct solution.
-
Izzie made a mistake going from step 3 to step 4:
- True; she didn’t correctly reverse the inequality.
-
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.
-
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