Let’s go through Izzie's work step by step to determine if she solved the inequality correctly.
Starting with the original inequality: \[ 3(t + 1) - 4t \geq -5 \]
Step 1: Simplifying the left-hand side \[ 3(t) + 3(1) - 4t \geq -5 \] This simplifies to: \[ 3t + 3 - 4t \geq -5 \]
Result: \[ -t + 3 \geq -5 \] (This matches Step 1 and Step 2 from Izzie's work)
Step 2: Rearranging the terms Subtract 3 from both sides: \[ -t \geq -5 - 3 \] \[ -t \geq -8 \]
Result: \[ -t \geq -8 \] (Here is where Izzie made a mistake from Step 2 to Step 3)
Step 3: Dividing by -1 When you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign: \[ t \leq 8 \]
This means that:
- Izzie's conclusion in Step 4 "t ≤ 2" is incorrect.
- The correct answer should actually be t ≤ 8.
So, let's analyze Izzie’s assertions:
- True: The correct answer should have been t ≤ 8 - This statement is true.
- Izzie made a mistake going from step 3 to step 4 - This is also true.
- Izzie made a mistake going from step 2 to step 3 - This is true as well (she incorrectly calculated the right-hand side).
- Izzie made no mistakes in solving this inequality - This is false.
- Izzie made a mistake going from step 1 to step 2 - This is false; her calculations from step 1 to step 2 were correct.
- Izzie should not have changed the inequality symbol from ≥ to ≤ - This statement is true, as she did not reverse it when she should have.
So the correct assessments of the statements are as follows:
- Statement 1: True
- Statement 2: True
- Statement 3: True
- Statement 4: False
- Statement 5: False
- Statement 6: True