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
Izzie made a mistake going from step 1 to step 2


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 2 to step 3


Izzie made no mistakes in solving this inequality


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

Answer 1

Let's review Izzie's steps to identify any mistakes in solving the inequality \(3(t+1)−4t≥−5\).

### Step-by-step analysis:

1. **Initial Inequality**:
\[
3(t+1) - 4t \geq -5
\]

2. **Step 1**: Distribute and simplify:
\[
3t + 3 - 4t \geq -5
\]
This is correct.

3. **Step 2**: Combine like terms:
\[
-t + 3 \geq -5
\]
This is also correct.

4. **Step 3**: Isolate \(t\):
\[
-t \geq -2
\]
This is also correct.

5. **Step 4**: Divide by -1 (which reverses the inequality sign):
\[
t \leq 2
\]
This step has a mistake because when you multiply or divide by a negative number, you must reverse the inequality sign.
The correct transformation should have been:
\[
t \leq 2
\]
So this step is correctly stated as leading to \(t \leq 2\).

### Conclusion based on analysis of each statement:

1. **Izzie made a mistake going from step 1 to step 2**:
- **False**

2. **Izzie made a mistake going from step 3 to step 4**:
- **False** (She did not make a mistake, but she should have acknowledged the reversal.)

3. **The correct answer should have been \(t \leq 8\)**:
- **False** (The correct answer should indeed be \(t \leq 2\).)

4. **Izzie made a mistake going from step 2 to step 3**:
- **False**

5. **Izzie made no mistakes in solving this inequality**:
- **False** (She needed to correctly note the reversal of the inequality sign when going from step 3 to step 4.)

6. **Izzie should not have changed the inequality symbol from ≥ to ≤**:
- **True** (While she correctly applied the rule, she forgot to recognize that the direction must reverse. Thus, she should not treat it as a standard sequence without noting the reciprocal.)

Overall, the mistake comes from understanding how to handle the inequality when dividing by a negative.