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.
| $|2 - 3x| = 7$ | (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) |
| $\displaystyle x = -\frac{5}{3}\ \text{ or } x = 3$ | (divide both sides of both equations by $\,-3\,$) |
| $3|-6x + 7| \le 9$ | (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) |
| $3|-6x + 7| \ge 9$ | (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) |
| $\displaystyle x\ge\frac{10}{6}\ \ \text{or}\ \ x\le \frac{4}{6}$ | (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) |
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.