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 that compare two expressions using symbols like <, >, \leq, or \geq, indicating that one quantity is less than, greater than, or not equal to another. Unlike equations, inequalities describe a range of possible solutions.

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 the inequality sign when multiplying or dividing both sides by a negative number
  • Confusing open circles (strict inequality <, >) with closed circles (inclusive \leq, \geq) on number lines
  • Treating inequalities exactly like equations — inequalities have a range of solutions, not just one value

Why This Formula Matters

Inequalities are essential for expressing constraints and boundaries in real life — from budgets and speed limits to engineering tolerances and scientific thresholds. They form the basis of optimization and linear programming.

Frequently Asked Questions

What is the Inequalities formula?

Mathematical statements that compare two expressions using symbols like <, >, \leq, or \geq, indicating that one quantity is less than, greater than, or not equal to another. Unlike equations, inequalities describe a range of possible solutions.

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?

Inequalities are essential for expressing constraints and boundaries in real life — from budgets and speed limits to engineering tolerances and scientific thresholds. They form the basis of optimization and linear programming.

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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