Laws of Monotonicity

The laws of monotonicity apply to both equations and inequalities.

First Law of Monotonicity
Second Law of Monotonicity for Equations
Second Law of Monotonicity for Inequalities

First Law of Monotonicity

An equation or inequality remains unchanged when the same number $ k $ is added to both sides. $$ a=b \Leftrightarrow a+k=b+k $$ $$ a<b \Leftrightarrow a+k<b+k $$ $$ a>b \Leftrightarrow a+k>b+k $$

For any real value of $ k $, the resulting equation or inequality is equivalent to the original one.

Example. This equality is true: $$ 2 = 2 $$ If we add 3 to both sides, the equality still holds: $$ 2+3 = 2+3 $$ $$ 5 = 5 $$ Example. This inequality is true: $$ 2<3 $$ If we add 1 to both sides, it remains true: $$ 2+1 < 3+1 $$ $$ 3<4 $$

Since $ k $ can also be negative ($ k<0 $), the first law of monotonicity also covers subtraction.

Example. This equality is true: $$ 5 = 5 $$ If we add -3 to both sides, the equality still holds: $$ 5+(-3) = 5+(-3) $$ $$ 5-3 = 5-3 $$ $$ 2 = 2 $$ Example. This inequality is true: $$ 8 > 5 $$ If we add -3 to both sides, the inequality remains valid: $$ 8+(-3) > 5+(-3) $$ $$ 8-3 > 5-3 $$ $$ 5 > 2 $$

When $ k=0 $, the equation or inequality is obviously unchanged, since zero is the additive identity.

Example. This equality is true: $$ 5 = 5 $$ If we add 0 to both sides, nothing changes: $$ 5+0 = 5+0 $$ $$ 5 = 5 $$

Second Law of Monotonicity for Equations

An equation remains true if both sides are multiplied by the same nonzero number $ k \ne 0 $. $$ a=b \Leftrightarrow a \cdot k=b \cdot k \ \ \ \text{with } k \ne 0 $$

For any nonzero value of $ k $, the resulting equation is equivalent to the original.

Example. This equality is true: $$ 2 = 2 $$ Multiplying both sides by 3 keeps it true: $$ 2 \cdot 3 = 2 \cdot 3 $$ $$ 6 = 6 $$

Since $ k $ can also be a reciprocal ($ 1/k $), this law applies to division as well:

$$ a=b \Leftrightarrow \frac{a}{k}=\frac{b}{k} \ \ \ \text{with } k \ne 0 $$

Example. This equality is true: $$ 8 = 8 $$ If we divide both sides by 2, it remains true: $$ \frac{8}{2} = \frac{8}{2} $$ $$ 4 = 4 $$

Why must zero be excluded?

Zero must be excluded because it’s the absorbing element of multiplication: multiplying by zero can turn a false statement into a true one.

Example. This statement is clearly false: $$ 2+1 = 10 $$ But if we multiply both sides by zero, it becomes trivially true: $$ (2+1) \cdot 0 = 10 \cdot 0 $$ $$ 0 = 0 $$

Moreover, in the case of division, using zero would lead to a division by zero - an operation that’s undefined.

Example. This statement is true: $$ 2+1=3 $$ But dividing both sides by zero would give: $$ \frac{2+1}{0}=\frac{3}{0} $$ which is undefined.

Second Law of Monotonicity for Inequalities

If $ k > 0 $, the direction of the inequality remains the same: $$ a<b \Longleftrightarrow a \cdot k<b \cdot k $$ $$ a>b \Longleftrightarrow a \cdot k>b \cdot k $$
If $ k < 0 $, the inequality sign must be reversed: $$ a<b \Longleftrightarrow a \cdot k>b \cdot k $$ $$ a>b \Longleftrightarrow a \cdot k<b \cdot k $$

For any nonzero $ k $, we obtain an equivalent inequality.

Example. This inequality is true: $$ 3<5 $$ If we multiply both sides by 2, it remains true and keeps the same direction: $$ 3 \cdot 2 < 5 \cdot 2 $$ $$ 6<10 $$ Example. This inequality is also true: $$ 3<5 $$ If we multiply both sides by -2, it still holds, but since the multiplier is negative, we must flip the inequality sign: $$ 3 \cdot (-2) > 5 \cdot (-2) $$ $$ -6 > -10 $$

The two laws of monotonicity also apply to subtraction and division - the inverse operations of addition and multiplication.

In such cases, they’re also referred to as the inverse laws of monotonicity, or simply as the cancellation laws.

And so on.