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 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 ≤

1 answer

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

The original inequality is: \[ 3(t + 1) - 4t \geq -5 \]

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

Step 2: Rearranging to isolate \( t \) \[ -t + 3 \geq -5 \] We subtract 3 from both sides: \[ -t \geq -8 \]

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: \[ t \leq 8 \] (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 \( t \leq 8 \):

    • 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 \( \geq \) to \( \leq \):

    • 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 \( t \leq 8 \): 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 \( \geq \) to \( \leq \): True