Absolute Value of a Real Number
In a parts factory in Monterrey, or in the high school lab, what matters is how far a measurement deviates from what was expected, not whether it came out too long or too short. That idea of "size of the deviation, without the sign" is the absolute value. At the end of the page you'll see how to use it to set margins of error and tolerances.
Definition and distance
Geometrically, $|x|$ is the distance from $x$ to the origin on the real number line. From this comes the key idea: $|a - b|$ is the distance between the points $a$ and $b$.
Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.
Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.
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.
Equations with absolute value
Solve the absolute value equation: $|3x| = 12$.
Solve the absolute value equation: $|3x| = 12$.
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.
Two absolute values and nested absolute values
A condition on distances can also be translated into absolute values: "$x$ is twice as far from $7$ as from $-2$" is written $|x - 7| = 2|x + 2|$, and it is solved with the two cases $x - 7 = 2(x+2)$ and $x - 7 = -2(x+2)$.
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: $|3x - 1| = |x + 5|$.
Find all real numbers $x$ such that their distance to point $7$ is exactly double their distance to point $-2$.
Find all real numbers $x$ such that their distance to point $7$ is exactly double their distance to point $-2$.
Solve the following equation containing nested absolute values: $||x - 2| - 5| = 3$.
Solve the following equation containing nested absolute values: $||x - 2| - 5| = 3$.
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.
Center, radius, and neighborhoods
The interval $[c - r, c + r]$ is written $|x - c| \le r$: $c$ is the center (midpoint) and $r$ the radius (half the length). For example, $(-8, 8)$ is $|x| < 8$, and $[3, 11]$ has center $c = \frac{3+11}{2} = 7$ and radius $r = \frac{11-3}{2} = 4$, so it is $|x - 7| \le 4$.
Neighborhoods will be useful later, when you study limits: "$x$ near $a$" means $x$ inside a neighborhood of $a$.
Tolerances
A tolerance says "the actual value $x$ must not differ from the ideal value $c$ by more than $r$": $|x - c| \le r$. If an ideal part measures 100 mm with a tolerance of 0.5 mm, it is acceptable if $|x - 100| \le 0.5$, that is $x \in [99.5, 100.5]$. It is rejected if $|x - 100| > 0.5$.
Write the interval $(-8, 8)$ using absolute value notation.
Write the interval $(-8, 8)$ using absolute value notation.
Express the closed interval $[3, 11]$ using an inequality of the form $|x - c| \le r$, determining the value of the center $c$ and the radius $r$.
Express the closed interval $[3, 11]$ using an inequality of the form $|x - c| \le r$, determining the value of the center $c$ and the radius $r$.
A steel bearing factory stipulates that the ideal length of a part is $250 \text{ mm}$. Quality control rejects any part whose length differs from the ideal by more than…
A steel bearing factory stipulates that the ideal length of a part is $250 \text{ mm}$. Quality control rejects any part whose length differs from the ideal by more than $1.2 \text{ mm}$. Express this condition using an absolute value inequality and determine the interval of valid lengths.
Applications: margins and deviations
When a problem talks about "differing," "deviating," or "being at a distance of," you translate it as $|x - c|$, with $c$ the reference value. If it says "no more than," use $\le$; if it says "more than" or "at least," use $>$ or $\ge$ as appropriate, and remember that this condition describes the zone outside the margin.
A bacterial culture in a biology laboratory must be kept at an exact temperature of $37^\circ\text{C}$. The experiment is ruined if the temperature fluctuates by…
A bacterial culture in a biology laboratory must be kept at an exact temperature of $37^\circ\text{C}$. The experiment is ruined if the temperature fluctuates by $2.5^\circ\text{C}$ or more. Write the condition for failure of the experiment using absolute value and solve it to find the critical temperature intervals.
The price of a stock on the stock market today is $1\,200$ pesos. A financial model predicts that next week's price $p$ will satisfy the relationship…
The price of a stock on the stock market today is $1\,200$ pesos. A financial model predicts that next week's price $p$ will satisfy the relationship $|p - 1\,200| \le 96$. What are the maximum and minimum prices envisioned by the prediction?
Fractions and parameters
In an inequality with $|x-1|$ in the denominator, first exclude the value that makes it zero. Since $|x-1| > 0$ for $x \ne 1$, you can multiply without changing the direction of the inequality.
For an equation with a parameter, such as $|x-4| + |x+1| = k$, think in terms of distances: $|x-4| + |x+1|$ is the sum of the distances from $x$ to $4$ and to $-1$. If $x$ is between them, the sum is $5$ (the distance between $4$ and $-1$); outside that segment, the sum is greater than $5$. Counting how many points give each value: if $k < 5$ there is no solution, if $k = 5$ there are infinitely many (all of $[-1, 4]$), and if $k > 5$ there are exactly two.
Solve the fractional inequality $\frac{3}{|x - 1|} \ge 2$.
Solve the fractional inequality $\frac{3}{|x - 1|} \ge 2$.
Discuss the number of solutions of the equation $|x - 4| + |x + 1| = k$ depending on the values of the real parameter $k$.
Discuss the number of solutions of the equation $|x - 4| + |x + 1| = k$ depending on the values of the real parameter $k$.
Removing the bars on an interval
If you know which interval $x$ lies in, you can remove the bars from each absolute value independently. You study the sign of what is inside: if it is positive (or zero), the bars are simply dropped; if it is negative, the sign of the entire inside expression is changed.
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 $(-2, 3)$.
Triangle inequality and error bounds
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.
The navigation system of a submarine requires its ideal depth $h$ (in meters, where $h > 0$) to remain in the range $|h - 250| \le 30$. However, strong ocean currents…
The navigation system of a submarine requires its ideal depth $h$ (in meters, where $h > 0$) to remain in the range $|h - 250| \le 30$. However, strong ocean currents introduce a disturbance, such that the submarine's actual depth $h_r$ satisfies $|h_r - h| \le 12$. Calculate the absolute interval of actual depths between which the submarine can move in the worst-case scenario.
In a machine learning model, the error $E$ between the prediction $y$ and the actual value $t$ is defined as $E = |y - t|$. If it is guaranteed by the design of the…
In a machine learning model, the error $E$ between the prediction $y$ and the actual value $t$ is defined as $E = |y - t|$. If it is guaranteed by the design of the model that $|y - 15| \le 1.5$ and $|t - 15| \le 2.5$, determine an upper bound for the maximum possible error $E$ using the properties of absolute value, and find the interval of possible values for the difference $y - t$.
Back to the beginning: a tolerance like "no more than 1.2 mm difference" from 250 mm is written $|x - 250| \le 1.2$, and its solution is the interval $[248.8, 251.2]$.