Q.$\dfrac{2x - 1}{3} \ge \dfrac{3x - 2}{4} - \dfrac{2 - x}{5}$

The key idea is to treat this as a linear inequality — clear denominators, simplify, and isolate $x$.

Multiply through by the LCM of 3, 4, and 5, which is 60:

This gives:

Expand and simplify each side:

Bring $x$-terms to one side and constants to the other: …

Let’s unpack the reasoning before we touch a single algebraic step. An inequality like this is just a statement that one expression is larger than or equal to another. The core skill is manipulating it without breaking the “balance” — especially remembering that multiplying or dividing by a negative flips the inequality sign. Here, we have fractions with denominators 3, 4, and 5. The cleanest path is to eliminate them all at once.

1. Find the least common multiple (LCM) of the denominators.

   The denominators are 3, 4, and 5. Their LCM is 60. Multiplying every term by 60 will clear all fractions in one go, turning the inequality into a simpler linear form.

2. Multiply the entire inequality by 60.

   Be careful: multiply each term separately, including the ones on the right-hand side that are subtracted.

3. Simplify each fraction.
   • First term: $60 \div 3 = 20$, so $20(2x - 1)$.
   • Second term: $60 \div 4 = 15$, so $15(3x - 2)$.
   • Third term: $60 \div 5 = 12$, so $12(2 - x)$. The inequality becomes:

4. Expand all brackets.

Notice the minus sign before $12(2 - x)$: it distributes as $-24 + 12x$ (since $-12 \times (-x) = +12x$). This is a common slip — keep the signs straight.

5. Combine like terms on the right-hand side. The $x$-terms: $45x + 12x = 57x$. The constants: $-30 - 24 = -54$. So:

6. Bring variable terms to one side, constants to the other. Subtract $57x$ from both sides:

Then add 20 to both sides:

7. Isolate $x$ by dividing by $-17$. Here’s the critical moment: dividing by a negative number reverses the inequality sign. $$$$