Linear Inequalities: Rules and Formulas

Equivalent Transformations
The Sign-Reversal Rule
Solving Procedure
Interval Notation Quick Reference
Solution Types
Formula Sheet
Practice Exercises
Related Articles

1. Equivalent Transformations

An equivalent transformation changes the form of an inequality without changing its solution set.

Transformation	Effect on Direction
Add the same number to both sides	no change
Subtract the same number from both sides	no change
Multiply both sides by a positive number	no change
Divide both sides by a positive number	no change
Multiply both sides by a negative number	reverses
Divide both sides by a negative number	reverses

2. The Sign-Reversal Rule

The single most important rule: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

Original	After multiplying/dividing by negative
$<$	$>$
$>$	$<$
$\leq$	$\geq$
$\geq$	$\leq$

Example:

- 3 x \leq 12 \Rightarrow x \geq - 4

We divided by $- 3$ (negative), so $\leq$ became $\geq$ .

3. Solving Procedure

Step 1: Expand parentheses (if any) Step 2: Eliminate fractions by multiplying by the LCD (if any) Step 3: Move all terms with

x

to one side, all constants to the other Step 4: Combine like terms Step 5: Divide by the coefficient of

x

(flip the sign if the coefficient is negative!) Step 6: Write the solution in inequality, interval, or number line form Step 7: Check by substituting a value from the solution set

4. Interval Notation Quick Reference

Inequality	Interval	Boundary
$x > a$	$(a, \infty)$	excluded
$x \geq a$	$[a, \infty)$	included
$x < a$	$(- \infty, a)$	excluded
$x \leq a$	$(- \infty, a]$	included

Remember: Infinity always gets a round bracket. The boundary gets a round bracket for strict ( $<$ , $>$ ) and a square bracket for non-strict ( $\leq$ , $\geq$ ).

5. Solution Types

Outcome	What Happens	Solution
Normal	$x$ compared to a number	an interval
No solution	false statement (e.g. $0 > 5$ )	$\emptyset$
All reals	true statement (e.g. $0 < 5$ )	$R$

6. Formula Sheet

Direct Solution Formulas

Inequality Type	Solution ( $a > 0$ )
$a x + b > c$	$x > \frac{c - b}{a}$
$a x + b < c$	$x < \frac{c - b}{a}$
$a x + b \geq c$	$x \geq \frac{c - b}{a}$
$a x + b \leq c$	$x \leq \frac{c - b}{a}$

When $a < 0$ (sign reverses!)

Inequality Type	Solution ( $a < 0$ )
$a x + b > c$	$x < \frac{c - b}{a}$
$a x + b < c$	$x > \frac{c - b}{a}$
$a x + b \geq c$	$x \leq \frac{c - b}{a}$
$a x + b \leq c$	$x \geq \frac{c - b}{a}$

Variable on Both Sides

For $a x + b > c x + d$ where $a \neq = c$ :

x > \frac{d - b}{a - c} if a - c > 0

x < \frac{d - b}{a - c} if a - c < 0

Practice Exercises

Inequalities - Basic - Apply the rules to simple problems
Inequalities - Both Sides - Variable on both sides
Inequalities - Mixed - Mixed practice of all types

Linear Inequalities - Comprehensive Guide - Full introduction with examples
Linear Inequalities - Simple Inequalities - Basic types step by step
Linear Inequalities - Intervals and Sets - Interval notation in detail
Linear Inequalities - Number Line - Graphical representation
Linear Inequalities - Special Cases - No solution and all real numbers

Linear Inequalities - Rules and Formulas

Linear Inequalities: Rules and Formulas

Table of Contents

1. Equivalent Transformations

2. The Sign-Reversal Rule

3. Solving Procedure

4. Interval Notation Quick Reference

5. Solution Types

6. Formula Sheet

Direct Solution Formulas

When $a < 0$ (sign reverses!)

Variable on Both Sides

Practice Exercises

Linear Inequalities - Rules and Formulas

Linear Inequalities: Rules and Formulas

Table of Contents

1. Equivalent Transformations

2. The Sign-Reversal Rule

3. Solving Procedure

4. Interval Notation Quick Reference

5. Solution Types

6. Formula Sheet

Direct Solution Formulas

When a<0 (sign reverses!)

Variable on Both Sides

Practice Exercises

Related Articles

When $a < 0$ (sign reverses!)