Question
14. Izzie is trying to solve this inequality: $3(t+1)-4 t \geq-5$ 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: $3 t+3-4 t \geq-5$ Step 2: $-t+3 \geq-5$ Step 3: $-t \geq-2$ Step 4: $t \leq 2$ (12 points) Izzie made a mistake going from step 3 to step 4 Izzie made a mistake going from step 1 to step 2 Izzie made a mistake going from step 2 to step 3 Tzie made no mistakes in solving this inequality Izzie should not have changed the inequality symbol from $\geq$ to $\leq$
Answers
Answered by
GPT-4o mini
Let's analyze Izzie's work step by step to identify if she made a mistake.
Starting with the original inequality:
\[
3(t+1) - 4t \geq -5
\]
**Step 1: Distributing the 3**
Izzie correctly distributed 3:
\[
3t + 3 - 4t \geq -5
\]
**Step 2: Combining like terms**
Next, she combined like terms (3t - 4t):
\[
-t + 3 \geq -5
\]
This step is correct.
**Step 3: Isolating t**
In this step, Izzie is isolating \( t \). She subtracts 3 from both sides:
\[
-t \geq -5 - 3
\]
\[
-t \geq -8
\]
Then there's a small error here. It appears she miscalculated:
If we calculate correctly, we get:
\[
-t \geq -8
\]
**Step 4: Multiplying by -1**
To isolate \( t \), Izzie will multiply both sides by -1. Remember that when we multiply or divide an inequality by a negative number, we must reverse the inequality sign. Thus:
\[
t \leq 8
\]
Now let's summarize her work:
1. Step 1 to Step 2: **Correct**
2. Step 2 to Step 3: **Incorrect** (She should have had \( -t \geq -8 \))
3. Step 3 to Step 4: **Incorrect** (She incorrectly reversed the inequality symbol and found \( t \leq 2 \) instead of the correct direction and number)
In conclusion, the primary mistake Izzie made is going from step 2 to step 3, as well as the mishandling of the inequality sign in step 4 (which we note also correlates since her arithmetic was wrong in step 3).
Thus, the correct statements are:
- **Izzie made a mistake going from step 2 to step 3.**
- **Izzie should not have changed the inequality symbol from \( \geq \) to \( \leq \).**
Therefore, Izzie's solution is incorrect, as shown. The correct solution is:
\[
t \geq 8
\]
So, among the provided statements, the most accurate one is:
**Izzie made a mistake going from step 2 to step 3**.
Starting with the original inequality:
\[
3(t+1) - 4t \geq -5
\]
**Step 1: Distributing the 3**
Izzie correctly distributed 3:
\[
3t + 3 - 4t \geq -5
\]
**Step 2: Combining like terms**
Next, she combined like terms (3t - 4t):
\[
-t + 3 \geq -5
\]
This step is correct.
**Step 3: Isolating t**
In this step, Izzie is isolating \( t \). She subtracts 3 from both sides:
\[
-t \geq -5 - 3
\]
\[
-t \geq -8
\]
Then there's a small error here. It appears she miscalculated:
If we calculate correctly, we get:
\[
-t \geq -8
\]
**Step 4: Multiplying by -1**
To isolate \( t \), Izzie will multiply both sides by -1. Remember that when we multiply or divide an inequality by a negative number, we must reverse the inequality sign. Thus:
\[
t \leq 8
\]
Now let's summarize her work:
1. Step 1 to Step 2: **Correct**
2. Step 2 to Step 3: **Incorrect** (She should have had \( -t \geq -8 \))
3. Step 3 to Step 4: **Incorrect** (She incorrectly reversed the inequality symbol and found \( t \leq 2 \) instead of the correct direction and number)
In conclusion, the primary mistake Izzie made is going from step 2 to step 3, as well as the mishandling of the inequality sign in step 4 (which we note also correlates since her arithmetic was wrong in step 3).
Thus, the correct statements are:
- **Izzie made a mistake going from step 2 to step 3.**
- **Izzie should not have changed the inequality symbol from \( \geq \) to \( \leq \).**
Therefore, Izzie's solution is incorrect, as shown. The correct solution is:
\[
t \geq 8
\]
So, among the provided statements, the most accurate one is:
**Izzie made a mistake going from step 2 to step 3**.