# Solving Absolute Value Sentences, All Types

This web exercise mixes up problems from the previous three web exercises:

- Solving Absolute Value Equations
- Solving Absolute Value Inequalities Involving ‘Less Than’
- Solving Absolute Value Inequalities Involving ‘Greater Than’

Visit these earlier sections for a thorough discussion of the concepts.

## Example: An Absolute Value Equation

$|2 - 3x| = 7$ | original equation |

$2-3x = \pm 7$ | check that $\,k\ge 0\,$; use the theorem |

$2-3x = 7$ or $\,2-3x = -7$ |
expand the plus/minus |

$-3x = 5$ 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 $\,-3\,$ |

It's a good idea to check your solutions:

$|2 - 3(-\frac{5}{3})|\ \overset{\text{?}}{=}\ 7$,
$|2 + 5| = 7$

Check!

$|2 - 3(3)|\ \overset{\text{?}}{=}\ 7$,
$|2 - 9| = 7$

Check!

## Example: An Absolute Value Inequality Involving ‘Less Than’

*all by itself*on one side of the sentence. 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 |

Check the ‘boundaries’ of the solution set:

$3|-6(\frac{2}{3}) + 7| = 3|-4 + 7| = 3|3| = 9$

Check!

$3|-6(\frac{5}{3}) + 7| = 3|-10 + 7| = 3|-3| = 9$

Check!

## Example: An Absolute Value Inequality Involving ‘Greater Than’

*all by itself*on one side of the sentence. 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 $\,3$ |

$-6x + 7 \le -3$ or $\,-6x + 7\ge 3$ |
check that $\,k \ge 0\,$; use the theorem |

$-6x\le -10$ 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 |

Check the ‘boundaries’ of the solution set:

$3|-6(\frac{2}{3}) + 7| = 3|-4 + 7| = 3|3| = 9$

Check!

$3|-6(\frac{5}{3}) + 7| = 3|-10 + 7| = 3|-3| = 9$

Check!

## Example: An Absolute Value Equation that is Always False

Can absolute value ever be negative?
No!
No matter *what* number you
substitute for $\,x\,,$
the left-hand side of the equation
will *always* be a number that is greater than or equal to zero.
Therefore, this sentence has no solutions.
It is always false.

## Example: An Absolute Value Inequality that is Always False

Can absolute value ever be negative?
No!
No matter *what* number you substitute
for $\,x\,,$
the left-hand side of the inequality
will *always* be a number that is greater than or equal to zero,
so it can't possibly be less than $\,-3\,.$
Therefore, this sentence has no solutions.
It is always false.

## Example: An Absolute Value Inequality that is Always True

No matter *what* number you
substitute for $\,x\,,$
the left-hand side of the inequality
will *always* be a number that is greater
than or equal to zero,
so it will *always* be greater
than $\,-3\,.$
Therefore, this sentence has all
real numbers as solutions.
It is always true.

## Concept Practice

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

For more advanced students, a graph is available. 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 to toggle.