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
The correct answer should have been t≤8


Izzie made a mistake going from step 3 to step 4


Izzie made a mistake going from step 2 to step 3


Izzie made no mistakes in solving this inequality


Izzie made a mistake going from step 1 to step 2


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 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:
1. **True: The correct answer should have been t ≤ 8** - This statement is true.
2. **Izzie made a mistake going from step 3 to step 4** - This is also true.
3. **Izzie made a mistake going from step 2 to step 3** - This is true as well (she incorrectly calculated the right-hand side).
4. **Izzie made no mistakes in solving this inequality** - This is false.
5. **Izzie made a mistake going from step 1 to step 2** - This is false; her calculations from step 1 to step 2 were correct.
6. **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