Inequalities Formula

The Formula

ax + b > c \implies x > \frac{c - b}{a} (flip sign if a < 0)

When to use: Instead of 'equals exactly,' it's 'at least,' 'at most,' or 'greater/less than.'

Quick Example

x + 3 > 7 \to x > 4 — any number greater than 4 works, such as 5, 10, or 100.

Notation

< less than, > greater than, \leq at most, \geq at least

What This Formula Means

Mathematical statements comparing expressions using <, >, \leq, or \geq.

Instead of 'equals exactly,' it's 'at least,' 'at most,' or 'greater/less than.'

Formal View

For a > 0: ax + b > c \iff x > \frac{c - b}{a}. For a < 0: ax + b > c \iff x < \frac{c - b}{a} (inequality reverses when multiplying by a negative).

Worked Examples

Example 1

easy
Solve 2x + 5 > 11.

Solution

  1. 1
    Subtract 5 from both sides: 2x > 6.
  2. 2
    Divide both sides by 2: x > 3.
  3. 3
    The solution is all values greater than 3.

Answer

x > 3
Solving inequalities follows the same steps as equations, with one key difference: multiplying or dividing by a negative number reverses the inequality sign.

Example 2

medium
Solve -3x + 4 \leq 13.

Common Mistakes

  • Forgetting to flip when multiplying by negative
  • Confusing \leq and <

Why This Formula Matters

Real-world constraints often involve ranges, not exact values.

Frequently Asked Questions

What is the Inequalities formula?

Mathematical statements comparing expressions using <, >, \leq, or \geq.

How do you use the Inequalities formula?

Instead of 'equals exactly,' it's 'at least,' 'at most,' or 'greater/less than.'

What do the symbols mean in the Inequalities formula?

< less than, > greater than, \leq at most, \geq at least

Why is the Inequalities formula important in Math?

Real-world constraints often involve ranges, not exact values.

What do students get wrong about Inequalities?

Always flip the inequality symbol when multiplying or dividing both sides by a negative number.

What should I learn before the Inequalities formula?

Before studying the Inequalities formula, you should understand: equations, integers.

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