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

Answer 1

Let's go through the steps to verify if Izzie's solution to the inequality \( 3(t + 1) - 4t \geq -5 \) is correct.

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

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

3. **Combine like terms in Step 2:**
\[
3t - 4t + 3 \geq -5
\]
This simplifies to:
\[
-t + 3 \geq -5
\]
This step is also correct.

4. **Isolate \( -t \) in Step 3:**
\[
-t \geq -5 - 3
\]
This simplifies to:
\[
-t \geq -8
\]
(Izzie wrote \(-t \geq -2\), which is incorrect.)

5. **Dividing by -1 in Step 4:**
Since we are dividing by a negative number, we must reverse the inequality:
\[
t \leq 8
\]

So, the correct solution to the inequality is \( t \leq 8 \).

**Conclusion:**
Izzie's conclusion that \( t \leq 2 \) is incorrect. The correct solution is \( t \leq 8 \).

Answer 2

the answer shoudlve have been t ≤ 8?

true or false

Answer 3

**True.**

The correct solution to the inequality \( 3(t + 1) - 4t \geq -5 \) simplifies to \( t \leq 8 \) after correctly following the steps in solving the inequality.

Answer 4

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

is it true or false

Answer 5

**True.**

Izzie correctly maintained the inequality symbol from \( \geq \) to \( \leq \) when isolating \( t \), provided that she divided both sides by -1. However, when you divide or multiply both sides of an inequality by a negative number, the inequality symbol *should* be reversed.

In her solution, she incorrectly simplified the expression at one point, making her conclusion incorrect. But regarding the inequality symbol itself: if she had correctly carried through the calculations and reversed it when dividing by -1, she would have had \( t \leq 8 \). Thus, if she changed it improperly without that context, the statement regarding changing the symbol is indeed true.

So, to clarify:
- The statement "Izzie should not have changed the inequality symbol from ≥ to ≤" is **true** because if she followed all steps correctly with the necessary reversals when dividing by a negative, she would maintain the integrity of the inequalities.

Answer 6

Izzie made a mistake going from step 1 to step 2?

true or false

Answer 7

**True.**

Izzie made a mistake going from Step 1 to Step 2.

In Step 1, she correctly simplified the left side to:
\[
3t + 3 - 4t \geq -5
\]
which simplifies to:
\[
-t + 3 \geq -5
\]

However, when she combined and isolated \( -t \) in Step 2, she incorrectly stated:
\[
-t + 3 \geq -5
\]
instead of correctly computing it to:
\[
-t \geq -8
\]

So, the conclusion that "Izzie made a mistake going from Step 1 to Step 2" is **true**.

Answer 8

Izzie made a mistake going from step 3 to step 4?

true or false

Answer 9

**True.**

Izzie made a mistake going from Step 3 to Step 4.

In Step 3, she correctly had:
\[
-t \geq -2
\]
However, when she moved to Step 4, she incorrectly stated that this implies \( t \leq 2 \).

The correct interpretation, after correctly isolating \( t \), should have resulted from reversing the inequality when dividing by -1:
\[
-t \geq -8 \implies t \leq 8
\]

Thus, since she did not correctly reverse the inequality or perform the necessary calculations in Step 4, the statement "Izzie made a mistake going from Step 3 to Step 4" is **true**.

Answer 10

Izzie made no mistakes in solving this inequality?

true or false

Answer 11

**False.**

Izzie made multiple mistakes while solving the inequality. Specifically:

1. She incorrectly simplified the inequality when trying to isolate \( -t \).
2. She made a mistake moving from Step 3 to Step 4 by failing to correctly reverse the inequality after multiplying or dividing by -1.

Thus, the statement "Izzie made no mistakes in solving this inequality" is **false**.