Absolute value of a real number
Absolute value appears whenever you talk about margin of error, manufacturing tolerance or distance: the sign of the deviation does not matter, only its size. Getting its rules right avoids sign errors that, in a component or a circuit, end up as defective parts.
Formal definition by cases
Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.
Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.
Determine which real values of $x$ satisfy the equation $|x| = -5$ and justify your answer.
Determine which real values of $x$ satisfy the equation $|x| = -5$ and justify your answer.
Equations with absolute value
Solve the absolute value equation: $|3x| = 12$.
Solve the absolute value equation: $|3x| = 12$.
Find all real values of $x$ that satisfy the equation with two absolute values: $|3x - 1| = |x + 5|$.
Find all real values of $x$ that satisfy the equation with two absolute values: $|3x - 1| = |x + 5|$.
Inequalities with absolute value
Express the inequality $|x| \le 5$ as an interval.
Express the inequality $|x| \le 5$ as an interval.
Express the inequality $|x + 2| > 4$ as a union of intervals.
Express the inequality $|x + 2| > 4$ as a union of intervals.
Solve the inequality $\left| \frac{2x - 3}{4} \right| \le 1$ and represent the solution as an interval.
Solve the inequality $\left| \frac{2x - 3}{4} \right| \le 1$ and represent the solution as an interval.
Distance and the triangle inequality
Calculate the distance on the number line between points $A = -6$ and $B = 11$ using the absolute value.
Calculate the distance on the number line between points $A = -6$ and $B = 11$ using the absolute value.
Prove, using the triangle inequality ($|x + y| \le |x| + |y|$), that for any real numbers $a$ and $b$, the inequality $|a - b| \ge ||a| - |b||$ holds.
Prove, using the triangle inequality ($|x + y| \le |x| + |y|$), that for any real numbers $a$ and $b$, the inequality $|a - b| \ge ||a| - |b||$ holds.
Functions defined with absolute value
Any expression with absolute values can be rewritten piecewise, without the bars, by studying the sign of each inner expression separately.
Simplify the algebraic expression $|x - 3| + |x + 2|$ knowing that the variable $x$ belongs to the interval $(-2, 3)$.
Simplify the algebraic expression $|x - 3| + |x + 2|$ knowing that the variable $x$ belongs to the interval $(-2, 3)$.
The following graph represents a function defined by the difference of two absolute values, $f(x) = |x - a| - |x - b|$ with $a < b$. By observing the vertices and…
The following graph represents a function defined by the difference of two absolute values, $f(x) = |x - a| - |x - b|$ with $a < b$. By observing the vertices and constant sections in the figure, determine the values of the parameters $a$ and $b$, and write the analytical expression of $f(x)$ defined piecewise without using absolute values.