Rules for Manipulating Inequalities

When working with algebraic inequalities, two foundational principles guide every valid transformation.

Changing the Sign of a Term
Reversing the Inequality

Changing the Sign of a Term

When a term is moved from one side of an inequality to the other, its sign must be reversed. This follows directly from adding or subtracting the same quantity from both sides.

Example

Consider the inequality:

$$ x + 1 > 3 $$

To move +1 to the right side, we use the first principle of equivalence, which states that adding or subtracting the same value from both sides preserves the inequality.

Subtract 1 from each side:

$$ x + 1 - 1 > 3 - 1 $$

$$ x > 3 - 1 $$

The term 1 is now on the right side with its sign reversed, and the inequality keeps the same direction.

Note. To move a negative term, add its opposite to both sides. For instance, in $$ x - 1 > 3 $$ adding +1 to each side gives $$ x - 1 + 1 > 3 + 1 $$ $$ x > 3 + 1 $$ The term -1 effectively appears on the right side as +1.

Reversing the Inequality

if the number is positive, the direction of the inequality stays the same
if the number is negative, the inequality must be reversed

Example 1

Consider the inequality:

$$ \frac{x+1}{2} > 3 $$

Multiply both sides by 2:

$$ \frac{x+1}{2} \cdot 2 > 3 \cdot 2 $$

Since 2 is positive, the direction of the inequality remains unchanged.

$$ x + 1 > 6 $$

Example 2

Take the same inequality:

$$ \frac{x+1}{2} > 3 $$

Multiply both sides by -2:

Because -2 is negative, the inequality must be reversed.

$$ \frac{x+1}{2} \cdot (-2) < 3 \cdot (-2) $$

$$ -x - 1 < -6 $$

Note. Division follows the same principle as multiplication: the inequality reverses only when the divisor is negative. For example, starting from $$ 2x > 4 $$ dividing both sides by -2 yields $$ \frac{2x}{-2} < \frac{4}{-2} $$ $$ -x < -2 $$ The direction flips because we divided by a negative number.

These rules underlie every valid manipulation of algebraic inequalities.