Q.Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.

Linear Inequalities: The Intuition First

You already know what an equation is: a statement that two things are exactly equal. $2x + 3 = 7$ says "twice something plus three is exactly seven." That's a tight, precise condition — only one number ($x = 2$) satisfies it.

Now imagine you loosen that condition. Instead of "exactly equal to 7," what if you said "less than 7"? Or "greater than or equal to 7"? That's an inequality. You're no longer looking for a single point; you're looking for a whole range of numbers.

The Four Symbols

There are only four inequality symbols. Memorise them once:

The "or equal to" versions ($\le$, $\ge$) include the boundary number itself. The strict versions ($<$, $>$) do not.

Solving Linear Inequalities: Almost Like Equations

A linear inequality looks just like a linear equation, but with an inequality sign instead of an equals sign. For example:

You solve it the same way you solve $2x + 3 = 7$ — with one critical difference.

The Golden Rule (and the only trap)

When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

Why? Think of the number line. $3 < 5$ is true. Multiply both sides by $-1$: $-3 < -5$? No — $-3$ is actually greater than $-5$ (because $-3$ is to the right on the number line). So the inequality flips: $-3 > -5$.

Example: Solve $2x + 3 < 7$

Step 1: Subtract 3 from both sides (no sign change — subtracting is always safe).

Step 2: Divide both sides by 2 (positive — no flip).

Answer: Any number less than 2 works. $x = 1.9$, $x = 0$, $x = -100$ — all satisfy the original inequality.

Example: Solve $-3x + 5 \ge 11$

Step 1: Subtract 5 from both sides.

Step 2: Divide both sides by $-3$ (negative — flip the sign).

Answer: $x$ must be less than or equal to $-2$.

Representing Solutions: The Number Line

The solution to an inequality is an interval (or union of intervals), not a single number. You can show it on a number line:

• Open circle at a number means that number is not included ($<$ or $>$).
• Closed circle means it is included ($\le$ or $\ge$).
• Shade the region that satisfies the inequality.

For $x < 2$: open circle at 2, shade everything to the left.

For $x \le -2$: closed circle at -2, shade everything to the left.

The Precise Definition

A linear inequality in one variable is any inequality that can be written in one of these four forms:

$ax + b < 0, \quad ax + b > 0, \quad ax + b \le 0, \quad ax + b \ge 0$ …