﻿ Absolute Value as Distance From Zero

# Absolute Value as Distance From Zero

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

Read through the following lessons: 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: $$\cssId{s13}{|x| = \text{the distance between } \,x\, \text{ and } \,0}$$ 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: $$\begin{gather} \cssId{s20}{|x| = k\ \text{ is equivalent to }\ x = \pm k} \\ \\ \cssId{s21}{|x| \lt k\ \ \text{ is equivalent to }\ -k \lt x \lt k} \\ \cssId{s22}{|x| \le k\ \ \text{ is equivalent to }\ -k \le x \le k} \\ \\ \cssId{s23}{|x| \gt k\ \text{ is equivalent to } (x\lt -k\ \text{ or }\ x\gt k)} \\ \cssId{s24}{|x| \ge k\ \text{ is equivalent to } (x\le -k\ \text{ or }\ x\ge k)} \end{gather}$$

## Examples

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
$$x = -\frac{5}{3}\ \text{ or } x = 3$$
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 inequality. 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
$$\frac{10}{6} \ge x \ge \frac{4}{6}$$
Divide all three parts by $\,-6\,$; change direction of inequality symbols
$$\frac{2}{3} \le x \le \frac{5}{3}$$
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 inequality. 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
$$x\ge\frac{10}{6}\ \ \text{or}\ \ x\le \frac{4}{6}$$
Divide by $\,-6\,$; change direction of inequality symbols
$$x\ge\frac{5}{3}\ \ \text{or}\ \ x\le \frac{2}{3}$$
Simplify fractions
$$x\le \frac{2}{3}\ \ \text{or}\ \ x\ge\frac{5}{3}$$
In the web exercise, the ‘less than’ part is always reported first

## Concept Practice

Solve the given absolute value sentence. Write the result in the most conventional way.

For more advanced students, a graph is displayed.

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. Click the ‘Show/Hide Graph’ button if you prefer not to see the graph.

Solve: