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14. 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 2 to step 3

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

Izzie made a mistake going from step 1 to step 2

Izzie made a mistake going from step 3 to step 4

Izzie made no mistakes in solving this inequality

The correct answer should have been t≤8

Question 16
15. Which graph represents the solutions to this inequality: 5n−3<3(n+1)
(2 points)
Responses

1 answer

Let's address the problems with Izzie's solution first.

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

    • In step 2, she correctly simplified to \(-t + 3 \geq -5\). However, when she subtracts 3 from both sides, the correction should be \(-t \geq -8\), not \(-t \geq -2\).

  2. False: Izzie should not have changed the inequality symbol from ≥ to ≤.

    • The inequality sign is not changed until you multiply or divide by a negative number; she did not do that at this step.

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

    • Her step from 3(t + 1) - 4t ≥ -5 to 3t + 3 - 4t ≥ -5 is actually correct.

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

    • As mentioned, the correct outcome of step 3 should lead to \(t \leq 8\) after multiplying by -1 and reversing the inequality (which actually should be -t ≥ -8).

  5. False: Izzie made no mistakes in solving this inequality.

    • She indeed made mistakes as detailed above.

  6. False: The correct answer should have been t ≤ 8.

    • As shown through analysis, the correct answer after solving \(5(t + 1) < 3(n + 1)\) simplifies to t ≤ 8, but her approach led her to the incorrect conclusion.

Question 15

Next, let’s solve the inequality given in Question 15:

5n - 3 < 3(n + 1)

  1. Distribute on the right side: \[5n - 3 < 3n + 3\]

  2. Subtract \(3n\) from both sides: \[2n - 3 < 3\]

  3. Add \(3\) to both sides: \[2n < 6\]

  4. Divide by \(2\): \[n < 3\]

The solution is \(n < 3\).

In terms of graphing, this would be represented on a number line as an open circle at \(3\) extending to the left, indicating all values less than \(3\).

Overall, for both parts, please make sure to confirm any graph posted corresponds with these conclusions.