Q.Solve the inequality $3x - 5 < 7$ for real $x$, and represent the solution set on the number line.

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:

where $a$ and $b$ are real numbers, and $a \neq 0$.

The solution set is the set of all real numbers $x$ that make the inequality true. It is always an interval (or a ray) on the real number line.

Two Quick Checks

1. Does $x = 3$ satisfy $2x - 5 > 0$?

   $2(3) - 5 = 6 - 5 = 1 > 0$ → Yes.

2. Does $x = 3$ satisfy $2x - 5 \ge 0$?

   $1 \ge 0$ → Yes (the boundary $x = 2.5$ also works here, but not in the strict version).

What Comes Next

Once you're comfortable with one variable, you'll extend to linear inequalities in two variables ($ax + by + c > 0$), where the solution set becomes a half-plane on the coordinate grid. But that's a step ahead — master the one-variable case first.

Final takeaway: Inequalities are just equations with a looser condition. Solve them the same way, but remember the flip rule when multiplying/dividing by a negative. The answer is always a range, not a point.

Linear inequalities in one variable form their own dedicated chapter in the NCERT Class 11 Mathematics syllabus, and "linear inequalities rules and sign flip when multiplying by negative" is a frequently searched revision topic for CBSE board preparation. This sign-flip rule is also a classic source of small but costly errors tested in "linear inequalities important questions" for competitive exams.