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.