6th Grade - Absolute Values

Introduction

• The symbol for absolute value is $\left| \right|$.
• The distance of a number (irrespective of its sign, positive or negative) from 0 on a number line is its absolute value. For e.g. $\left|-3\right| = 3$, the distance of -3 from 0 on a number line is 3, or $\left|+10\right| = 10$, because the distance of 10 from 0 on a number line is 10.
• The absolute value of any number (+ve or -ve) is always positive.

Examples

• $\left|-13\right| = 13$
• $\left|+14\right| = 14$
• $\left|-21\right| = 21$
• $\left|12 - 3\right| = \left|9\right| = 9$
• $\left|17 + 2\right| = \left|19\right| = 19$

Examples with the negative sign outside the absolute value symbol:

• $-\left|-11\right| = -11$
• $-\left|-23\right| = -23$
• $-\left|-45\right| = -45$
• $-\left|12 + 7\right| = -19$
• $-\left|-11-8\right| = -\left|-19\right| = -19$

Cheat Sheet

• The absolute value of $\left|0\right| = 0$, because the distance of 0 from 0 is 0.
• Nested absolute value is possible, for e.g.$\left|\left|-3\right|+\left|-2\right|\right|$ . In such a case, work on the outside absolute value first, and then move inside. e.g. $$\begin{gathered} \left|\left|-3\right|+\left|-2\right|\right| \\ = \left|-3\right| +\left|-2\right| \\ = 3 + 2 \\ =5 \end{gathered}$$
• The expression inside the absolute value is possible, for e.g. $\left|-2x+4\right| = 8$.

Blunder Area

• If the negative sign is outside the absolute value symbol, the answer/value will take the negative sign.     

For example:  $-\left|-6\right|= -6$

Because when we remove the absolute value symbol, the negative sign inside the symbol drops by itself, but the negative sign outside the symbol will remain there.