Q.Solve the following inequality and write the solution set using interval notation: −9 < 2x + 7 ≤ 19.

Solving an inequality means finding the full set of values that make it true, and interval notation is the standard way to express that solution set compactly. Two ground rules govern how algebraic operations affect an inequality: adding, subtracting, multiplying, or dividing by a POSITIVE number preserves the direction of the inequality; but multiplying or dividing by a NEGATIVE number REVERSES it. A compound linear inequality like −9 < 2x+7 ≤ 19 is solved by applying the same operation (subtract 7, then divide by 2) to all three parts simultaneously, carefully tracking which end stays strict and which stays inclusive. A quadratic or rational inequality (like x²−x > 20, or 2x/(x−4) ≤ 5) requires a different technique — sign analysis: move every term to one side (so the other side is 0), factor or combine into a single fraction, mark the critical points where the numerator or denominator is zero, and test one representative value from each region these points create on the number line to determine the sign of the expression there. A point that makes a DENOMINATOR zero must …