﻿ Absolute Value as Distance From Zero

# ABSOLUTE VALUE AS DISTANCE FROM ZERO

• PRACTICE (online exercises and printable worksheets)

A solid understanding of absolute value is vital for success in Precalculus and Calculus.

the first few should be quick-and-easy, but it's important to make sure your foundational concepts are sound.
Be sure to click-click-click the web exercises in each section to check your understanding!
The lessons will open in a new tab/window.

If you're in a hurry, here are the key concepts and a few examples.
The web exercises on this page are a duplicate of those in Solving Absolute Value Sentences, All Types.

DEFINITION absolute value (geometric definition)
Let $\,x\,$ be a real number. Then: $$|x| = \text{the distance between } \,x\, \text{ and } \,0$$ The symbol $\,|x|\,$ is read as the absolute value of $\,x\,$.
THEOREM solving absolute value sentences
Let $\,x\in\mathbb{R}\,$, and let $\,k\ge 0\,$.   Then, $$\begin{gather} |x| = k\ \ \text{ is equivalent to }\ \ x = \pm k \\ \\ |x| \lt k\ \ \ \text{ is equivalent to }\ \ -k \lt x \lt k \\ |x| \le k\ \ \ \text{ is equivalent to }\ \ -k \le x \le k \\ \\ |x| \gt k\ \ \ \text{ is equivalent to }\ \ x\lt -k\ \ \text{ or }\ \ x\gt k \\ |x| \ge k\ \ \ \text{ is equivalent to }\ \ x\le -k\ \ \text{ or }\ \ x\ge k \end{gather}$$
EXAMPLE (an absolute value equation):
Solve: $|2 - 3x| = 7$
Solution:
 $|2 - 3x| = 7$ (original equation) $2-3x = \pm 7$ (check that $\,k\ge 0\,$; use the theorem) $2-3x = 7\ \text{ or }\ 2-3x = -7$ (expand the plus/minus) $-3x = 5\ \text{ or }\ -3x = -9$ (subtract $\,2\,$ from both sides of both equations) $\displaystyle x = -\frac{5}{3}\ \text{ or } x = 3$ (divide both sides of both equations by $\,-3\,$)
EXAMPLE (an absolute value inequality involving ‘less than’):
Solve: $3|-6x + 7| \le 9$
Solution:
To use the theorem, you must have the absolute value all by itself on one side of the equation.
Thus, your first job is to isolate the absolute value:
 $3|-6x + 7| \le 9$ (original sentence) $|-6x + 7| \le 3$ (divide both sides by $\,3$) $-3 \le -6x + 7 \le 3$ (check that $\,k \ge 0\,$; use the theorem) $-10 \le -6x \le -4$ (subtract $\,7\,$ from all three parts of the compound inequality) $\displaystyle \frac{10}{6} \ge x \ge \frac{4}{6}$ (divide all three parts by $\,-6\,$; change direction of inequality symbols) $\displaystyle \frac{2}{3} \le x \le \frac{5}{3}$ (simplify fractions; write in the conventional way)
EXAMPLE (an absolute value inequality involving ‘greater than’):
Solve: $3|-6x + 7| \ge 9$
Solution:
To use the theorem, you must have the absolute value all by itself on one side of the equation.
Thus, your first job is to isolate the absolute value:
 $3|-6x + 7| \ge 9$ (original sentence) $|-6x + 7| \ge 3$ (divide both sides by $\,3$) $-6x + 7 \le -3\ \ \text{or}\ \ -6x + 7\ge 3$ (check that $\,k \ge 0\,$; use the theorem) $-6x\le -10\ \ \text{or}\ \ -6x\ge -4$ (subtract $\,7\,$ from both sides of both subsentences) $\displaystyle x\ge\frac{10}{6}\ \ \text{or}\ \ x\le \frac{4}{6}$ (divide by $\,-6\,$; change direction of inequality symbols) $\displaystyle x\ge\frac{5}{3}\ \ \text{or}\ \ x\le \frac{2}{3}$ (simplify fractions) $\displaystyle x\le \frac{2}{3}\ \ \text{or}\ \ x\ge\frac{5}{3}$ (in the web exercise, the ‘less than’ part is always reported first)
Master the ideas from this section

When you're done practicing, move on to:
Absolute Value as Distance Between Two Numbers
For example, the inequality $\,|2 - 3x| \lt 7\,$
is optionally accompanied by the graph of $\,y = |2 - 3x|\,$ (the left side of the inequality, dashed green)
and the graph of $\,y = 7\,$ (the right side of the inequality, solid purple).
In this example, you are finding the values of $\,x\,$ where the green graph lies below the purple graph.