- PRACTICE (online exercises and printable worksheets)

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:
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$$
|x| = \text{the distance between } \,x\, \text{ and } \,0
$$
The symbol $\,|x|\,$ is read as *the absolute value of* $\,x\,$.

THEOREM
solving absolute value sentences

Let
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$\,x\in\mathbb{R}\,$,
and let
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$\,k\ge 0\,$.
Then,
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$$
\begin{gather}
|x| = k\ \ \text{ is equivalent to }\ \ x = \pm k \\
\\
|x| \lt k\ \ \ \text{ is equivalent to }\ \ -k \lt x \lt k \\
|x| \le k\ \ \ \text{ is equivalent to }\ \ -k \le x \le k \\
\\
|x| \gt k\ \ \ \text{ is equivalent to }\ \ x\lt -k\ \ \text{ or }\ \ x\gt k \\
|x| \ge k\ \ \ \text{ is equivalent to }\ \ x\le -k\ \ \text{ or }\ \ x\ge k
\end{gather}
$$

EXAMPLE (an absolute value equation):

Solve:
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$|2 - 3x| = 7$

Solution:

[beautiful math coming... please be patient] $|2 - 3x| = 7$ | (original equation) |

[beautiful math coming... please be patient] $2-3x = \pm 7$ | (check that $\,k\ge 0\,$; use the theorem) |

[beautiful math coming... please be patient] $2-3x = 7\ \text{ or }\ 2-3x = -7$ | (expand the plus/minus) |

[beautiful math coming... please be patient] $-3x = 5\ \text{ or }\ -3x = -9$ | (subtract $\,2\,$ from both sides of both equations) |

[beautiful math coming... please be patient] $\displaystyle x = -\frac{5}{3}\ \text{ or } x = 3$ | (divide both sides of both equations by $\,-3\,$) |

EXAMPLE (an absolute value inequality involving ‘less than’):

Solve:
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$3|-6x + 7| \le 9$

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*:

To use the theorem, you must have the absolute value

Thus, your first job is to

[beautiful math coming... please be patient] $3|-6x + 7| \le 9$ | (original sentence) |

[beautiful math coming... please be patient] $|-6x + 7| \le 3$ | (divide both sides by $\,3$) |

[beautiful math coming... please be patient] $-3 \le -6x + 7 \le 3$ | (check that $\,k \ge 0\,$; use the theorem) |

[beautiful math coming... please be patient] $-10 \le -6x \le -4$ | (subtract $\,7\,$ from all three parts of the compound inequality) |

[beautiful math coming... please be patient] $\displaystyle \frac{10}{6} \ge x \ge \frac{4}{6}$ | (divide all three parts by $\,-6\,$; change direction of inequality symbols) |

[beautiful math coming... please be patient] $\displaystyle \frac{2}{3} \le x \le \frac{5}{3}$ | (simplify fractions; write in the conventional way) |

EXAMPLE (an absolute value inequality involving ‘greater than’):

Solve:
[beautiful math coming... please be patient]
$3|-6x + 7| \ge 9$

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*:

To use the theorem, you must have the absolute value

Thus, your first job is to

[beautiful math coming... please be patient] $3|-6x + 7| \ge 9$ | (original sentence) |

[beautiful math coming... please be patient] $|-6x + 7| \ge 3$ | (divide both sides by $\,3$) |

[beautiful math coming... please be patient] $-6x + 7 \le -3\ \ \text{or}\ \ -6x + 7\ge 3$ | (check that $\,k \ge 0\,$; use the theorem) |

[beautiful math coming... please be patient] $-6x\le -10\ \ \text{or}\ \ -6x\ge -4$ | (subtract $\,7\,$ from both sides of both subsentences) |

[beautiful math coming... please be patient] $\displaystyle x\ge\frac{10}{6}\ \ \text{or}\ \ x\le \frac{4}{6}$ | (divide by $\,-6\,$; change direction of inequality symbols) |

[beautiful math coming... please be patient] $\displaystyle x\ge\frac{5}{3}\ \ \text{or}\ \ x\le \frac{2}{3}$ | (simplify fractions) |

[beautiful math coming... please be patient] $\displaystyle x\le \frac{2}{3}\ \ \text{or}\ \ x\ge\frac{5}{3}$ | (in the web exercise, the ‘less than’ part is always reported first) |

Master the ideas from this section

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Absolute Value as Distance Between Two Numbers

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Absolute Value as Distance Between Two Numbers

On this exercise, you will not key in your answer.

However, you can check to see if your answer is correct.

However, you can check to see if your answer is correct.

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.