Solve the inequality: $\frac{-m}{2} - \frac{5}{4} \geq \frac{5m}{12} - \frac{7}{6}$

Step 1: Clear the fractions. The denominators are 2, 4, 12 and 6, so the LCM is 12. Multiply every term by 12:

12(−m/2) − 12(5/4) ≥ 12(5m/12) − 12(7/6)

−6m − 15 ≥ 5m − 14

Step 2: Collect like terms. Add 6m to both sides and add 14 to both sides:

−15 + 14 ≥ 5m + 6m

−1 ≥ 11m

Step 3: Divide by 11. Dividing by a positive number does not change the direction of the inequality:

−1/11 ≥ m, which means m ≤ −1/11

Check: Take m = −1, which satisfies m ≤ −1/11.

• Left side: 1/2 − 5/4 = −3/4
• Right side: −5/12 − 7/6 = −5/12 − 14/12 = −19/12
• Is −3/4 ≥ −19/12? Since −9/12 ≥ −19/12, yes.

Tip: Always multiply by the LCM first to remove the fractions. The inequality sign only flips when you multiply or divide by a negative number. Here we divided by +11, so it stayed the same. Also, "−1 ≥ 11m" is the same as "m ≤ −1/11", so read it from the variable's side.