Solve $\frac{y+2}{4}$ - $\frac{y-1}{3}$ > 1

Analyze the question: We need to find the values of y that satisfy the inequality (y+2)/4 − (y−1)/3 > 1.

Concept: To remove fractions, multiply both sides by the LCM of the denominators. Since the LCM is positive, the inequality sign stays the same. But if you multiply or divide by a negative number, the sign must flip.

Solution:

1. The LCM of 4 and 3 is 12. Multiply every term by 12:

12(y+2)/4 − 12(y−1)/3 > 12(1)

2. Simplify:

3(y+2) − 4(y−1) > 12

3. Expand the brackets (be careful with the minus sign in front of the second bracket):

3y + 6 − 4y + 4 > 12

4. Collect like terms:

−y + 10 > 12

5. Subtract 10 from both sides:

−y > 2

6. Divide by −1, which flips the sign:

y < −2

Check: Try y = −3 (which satisfies y < −2): (−3+2)/4 − (−3−1)/3 = −1/4 + 4/3 = (−3 + 16)/12 = 13/12, which is greater than 1. ✓

Try y = 0 (which does not satisfy it): 2/4 + 1/3 = 5/6, which is less than 1. ✓ (so it is correctly excluded)

Answer: y < −2, which is option B.

Tip: The most common mistakes here are forgetting to distribute the negative sign across (y−1) and forgetting to flip the inequality sign when dividing by a negative number. Always test a value from your answer to confirm.