Izzie is trying to solve this inequality: 3(t+1)−4t≥−5

Question

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 no mistakes in solving this inequality

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 should not have changed the inequality symbol from ≥
to ≤

Answer 1

Let's go through Izzie's steps to verify whether she solved the inequality correctly:

The original inequality is: $$3(t+1)-4t\ge -5$$

Step 1: Expand and simplify $$3t+3-4t\ge -5$$ This simplifies to: $$-t+3\ge -5$$ (So Step 1 to Step 2 is correct.)

Step 2: Rearranging to isolate $t$ $$-t+3\ge -5$$ We subtract 3 from both sides: $$-t\ge -8$$

Step 3: Dividing by -1 (this reverses the inequality) When you divide or multiply an inequality by a negative number, you must reverse the inequality symbol: $$t\le 8$$ (So Step 3 to Step 4 is not correct, as Izzie didn't change the inequality direction when multiplying by -1.)

Now, let's evaluate the statements:

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

   • False; she correctly simplified in step 1.

2. Izzie made no mistakes in solving this inequality:

   • False; she made a mistake in step 3.

3. The correct answer should have been $t\le 8$:

   • True; this is the correct solution.

4. Izzie made a mistake going from step 3 to step 4:

   • True; she didn’t correctly reverse the inequality.

5. Izzie made a mistake going from step 2 to step 3:

   • False; the transition was correct, but she mismanaged the inequality in the next step.

6. Izzie should not have changed the inequality symbol from $\ge$ to $\le$:

   • True; she should have reversed the direction of the inequality when dividing by -1.

So, the final answers to the statements are:

• Izzie made a mistake going from step 1 to step 2: False
• Izzie made no mistakes in solving this inequality: False
• The correct answer should have been $t\le 8$: True
• Izzie made a mistake going from step 3 to step 4: True
• Izzie made a mistake going from step 2 to step 3: False
• Izzie should not have changed the inequality symbol from $\ge$ to $\le$: True