14. 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

Answer 1

Let's go through the steps of Izzie's work to check if the solution is correct.

We start with the inequality: \[ 3(t + 1) - 4t \geq -5. \]

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

Step 2: Move the constant to the right Now, we'll isolate the variable: \[ -t + 3 \geq -5. \] Subtracting 3 from both sides gives: \[ -t \geq -8. \] This step is incorrect; it should have been: \[ -t \geq -5 - 3, \]
which simplifies to: \[ -t \geq -8. \]

Step 3: Solve for \( t \) Now, we flip the inequality when we multiply or divide by a negative number: \[ t \leq 8. \] Again, this is where Izzie made a mistake. The correct inequality should be: \[ t \leq 8. \]

Therefore, the correct solution to the inequality is: \[ t \leq 8. \]

In her last step, Izzie wrote \( t \leq 2 \), which is incorrect. The correct solution is \( t \leq 8 \).