Solving Absolute Value Sentences, All Types

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

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

THEOREM solving absolute value sentences

Let $\,x\in\mathbb{R}\,,$ and let $\,k\ge 0\,.$ Then: $$ \begin{gather} \cssId{s13}{|x| = k\ \ \text{ is equivalent to }\ \ x = \pm k} \\ \\ \cssId{s14}{|x| \lt k\ \ \ \text{ is equivalent to }\ \ -k \lt x \lt k} \\ \cssId{s15}{|x| \le k\ \ \ \text{ is equivalent to }\ \ -k \le x \le k} \\ \\ \cssId{s16}{|x| \gt k\ \ \ \text{ is equivalent to }\ \ x\lt -k\ \ \text{ or }\ \ x\gt k} \\ \cssId{s17}{|x| \ge k\ \ \ \text{ is equivalent to }\ \ x\le -k\ \ \text{ or }\ \ x\ge k} \end{gather} $$

Example: An Absolute Value Equation

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

Solution: Write a nice, clean list of equivalent sentences:

$\|2 - 3x\| = 7$	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’

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 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’

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 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

Solve: $|3x + 1| = -5$

Solution: The theorem can't be used here, since $\,k\,$ is negative. In this case, you need to stop and think.

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

Solve: $|5 - 2x| \lt -3$

Solution: The theorem can't be used here, since $\,k\,$ is negative. In this case, you need to stop and think.

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

Solve: $|5 - 2x| \gt -3$

Solution: The theorem can't be used here, since $\,k\,$ is negative. In this case, you need to stop and think.

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.