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
True or False
Izzie made a mistake going from step 2 to step 3
Izzie made a mistake going from step 3 to step 4
Izzie should not have changed the inequality symbol from ≥
to ≤
Izzie made a mistake going from step 1 to step 2
The correct answer should have been t≤8
Izzie made no mistakes in solving this inequality
To solve the inequality correctly, let's go through each step:
Step 1: Distribute the 3 to both terms inside the parentheses:
3(t+1) - 4t ≥ -5
3t + 3 - 4t ≥ -5
Step 2: Combine like terms:
-t + 3 ≥ -5
So far, so good. However, Izzie made a mistake in going from step 2 to step 3. When multiplying or dividing both sides of an inequality by a negative number, the inequality symbol should be reversed.
Step 3: Multiply both sides by -1 (and reverse the inequality):
-t ≤ -2
Now, Izzie made another mistake by changing the inequality symbol from ≤ to ≥. It should remain as ≤ since we multiplied both sides by a negative number.
So, the correct answer should be: -t ≤ -2
Finally, solving for t, we need to multiply both sides by -1 again. Remember to reverse the inequality symbol again since we are multiplying by a negative number.
Step 4: Multiply both sides by -1 (and reverse the inequality):
t ≥ 2
Therefore, the correct answer is t ≥ 2.
Izzie made mistakes going from step 2 to step 3 and in changing the inequality symbol from ≥ to ≤.
Its true or false
False. Izzie made a mistake going from step 2 to step 3 and also made a mistake in changing the inequality symbol from ≥ to ≤.
Izzie made a mistake going from step 3 to step 4.
In step 3, -t ≥ -2 is correct. However, to isolate t, she needs to divide both sides of the inequality by -1. When dividing by a negative number, the inequality direction should be flipped. Therefore, the correct step 4 would be t ≤ 2, not t ≥ 2.
So, the correct answer is: Izzie made a mistake going from step 3 to step 4.
Izzie made a mistake going from step 2 to step 3. Let's go through each step to understand what she did correctly and where the mistake occurred:
Step 1: Simplify the left side of the inequality:
3(t + 1) - 4t ≥ -5
Distribute 3 to (t + 1):
3t + 3 - 4t ≥ -5
Combine like terms on the left side:
-t + 3 ≥ -5
Step 2: Isolate the variable by subtracting 3 from both sides:
-t + 3 - 3 ≥ -5 - 3
Simplify:
-t ≥ -8
At this point, Izzie made a mistake. She incorrectly changed the inequality symbol from ≥ to ≤ in Step 3.
The correct step 3 would be to multiply both sides of the inequality by -1 and change the direction of the inequality because multiplying or dividing by a negative number reverses the inequality:
(-1)(-t) ≤ (-1)(-8)
Simplify:
t ≤ 8
Therefore, Izzie incorrectly changed the inequality symbol from ≥ to ≤ in step 3.
The correct answer is: Izzie made a mistake going from step 2 to step 3.