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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 $\,x\,.$
THEOREMsolving absolute value sentences
Let $\,x\in\mathbb{R}\,,$ and let $\,k\ge 0\,.$ Then:
$$
\begin{gather}
\cssId{s20}{|x| = k\ \text{ is equivalent to }\ x = \pm k} \\
\\
\cssId{s21}{|x| \lt k\ \ \text{ is equivalent to }\ -k \lt x \lt k} \\
\cssId{s22}{|x| \le k\ \ \text{ is equivalent to }\ -k \le x \le k} \\
\\
\cssId{s23}{|x| \gt k\ \text{ is equivalent to } (x\lt -k\ \text{ or }\ x\gt k)} \\
\cssId{s24}{|x| \ge k\ \text{ is equivalent to } (x\le -k\ \text{ or }\ x\ge k)}
\end{gather}
$$
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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
Choose a specific problem type, or click ‘New problem’ for a random question.
Think about your answer.
Click ‘Check your answer’ to check!
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.
Absolute Value as Distance From Zero
