Rules for solving an inequality

Rule 1: adding the same number to both sides, or subtracting the same number from both sides, keeps the inequality true with the same sign.

Rule 2: multiplying or dividing both sides by the same positive number keeps the sign. Multiplying or dividing both sides by a negative number reverses the sign: < becomes >, ≤ becomes ≥, and so on.

Why the reversal: 5 > 3, but −5 < −3. Multiplying by −1 reflects numbers through 0 on the number line, so their order swaps. Similarly −6 < −4, yet (−6)(−2) = 12 > 8 = (−4)(−2).

Worked example: 7x − 4 < 4x + 8. Subtract 4x: 3x − 4 < 8. Add 4: 3x < 12. Divide by 3 (positive): x < 4. Check: x = 3 gives 17 < 20 (true) and x = 5 gives 31 < 28 (false).

Worked example with a reversal: 2x + 5 > 5x − 10. Subtract 5x: −3x + 5 > −10. Subtract 5: −3x > −15. Divide by −3 and reverse: x < 5. Check: x = 4 gives 13 > 10 (true).

Worked example: −3x ≥ 12. Divide by −3 and reverse: x ≤ −4. Check: x = −5 gives 15 ≥ 12 (true) and x = 0 gives 0 ≥ 12 (false).

One safe habit: gather the x-terms on the side where their coefficient stays positive. For 2x + 5 > 5x − 10, moving 2x to the right gives 15 > 3x, so 5 > x with no reversal needed.

Never multiply or divide by an expression whose sign is unknown, such as x itself, without splitting into cases; the direction of the sign would be unknown.