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.
Real life is full of inequalities: "You need at least 60% to pass" (marks≥60), "The bus can carry at most 50 people" (passengers≤50), "Profit must be more than zero" (P>0). Equations are rare; inequalities are everywhere.
The Four Symbols
There are only four inequality symbols. Memorise them once:
| Symbol | Meaning | Example | Reads as |
|---|
| < | less than | x<5 | x is less than 5 |
| > | greater than | x>5 | x is greater than 5 |
| ≤ | less than or equal to | x≤5 | x is at most 5 |
| ≥ | greater than or equal to | x≥5 | x is at least 5 |
The "or equal to" versions (≤, ≥) 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.
This is the single most common mistake students make. If you multiply or divide by a negative, flip the sign. If you multiply/divide by a positive, leave it alone.
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≥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 (≤ or ≥).
- Shade the region that satisfies the inequality.
For x<2: open circle at 2, shade everything to the left.
For x≤−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,ax+b>0,ax+b≤0,ax+b≥0 …