Your thermostat is set to $70^\circ\text{F}$. The heater or the AC kicks in only when the room is more than $2$ degrees away from $70$, whether it is too warm or too cold. How can you say "more than 2 degrees away" in one short statement, and which temperatures trigger the system? Absolute value is the tool for exactly this: it measures how far something is from a target and ignores the direction.
Definition and distance
The absolute value of a number is its distance from $0$ on the real line. Distance is never negative, so the sign of the number does not matter, only its size.
Some problems call this the modulus of a number. It is the same thing.
The same idea works between any two points. The distance between $a$ and $b$ on the real line is $|a - b|$ (or $|b - a|$, it is the same number).
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Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.
Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.
Problems about distances turn into these equations. A statement like "$x$ is $3$ times as far from $1$ as from $5$" means you write each distance as $|x - p|$ for the point $p$ and set up the equation.
For nested bars, work from the outside in.
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Find all the real values of $x$ that satisfy the equation with two moduli: $|3x - 1| = |x + 5|$.
Find all the real values of $x$ that satisfy the equation with two moduli: $|3x - 1| = |x + 5|$.
Engineers, nurses, and coaches all use the same pattern: a target value and an allowed error.
Back to the thermostat: the system runs when $|T - 70| > 2$. That means $T < 68$ or $T > 72$. It stays off for $68 \le T \le 72$.
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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 a modulus inequality and determine the interval of valid lengths.
A bacteria 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 bacteria 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 failure condition of the experiment using a modulus and solve it to find the critical temperature intervals.
The price of a stock market share today is $\$60$. A financial model predicts that next week's price $p$ will satisfy the relationship $|p - 60| \le 4.8…
The price of a stock market share today is $\$60$. A financial model predicts that next week's price $p$ will satisfy the relationship $|p - 60| \le 4.8$. What are the maximum and minimum prices contemplated by the prediction?
The same two rules work when $A$ is a more complicated expression. You just end up solving ordinary inequalities afterward.
When the absolute value is in a denominator, first note what is not allowed. Then multiply by the absolute value, which is positive, so the inequality does not flip.
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Solve the quadratic modulus inequality: $|x^2 - 2x| < 3$.
Solve the quadratic modulus inequality: $|x^2 - 2x| < 3$.
A very useful consequence: if two errors are bounded, their sum is bounded by the sum of the bounds. Write $A = (A - B) + (B - C)$ and apply the inequality.
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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.
A submarine's navigation system requires its ideal depth $h$ (in feet, where $h > 0$) to be maintained in the range $|h - 250| \le 30$. However, strong ocean currents…
A submarine's navigation system requires its ideal depth $h$ (in feet, where $h > 0$) to be maintained in the range $|h - 250| \le 30$. However, strong ocean currents introduce a disturbance, so 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 model's design…
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 model's design 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 the modulus, and find the interval of possible values for the difference $y - t$.
To remove the bars, find where each expression inside changes sign. Those points split the number line into pieces. On each piece, every absolute value is just $+(\ldots)$ or $-(\ldots)$.
Notice the constant $3$ is the distance between $-1$ and $2$. The sum $|x - 2| + |x + 1|$ is the total distance from $x$ to the two points, and between them that total never changes. Outside, it grows.
Reading the graph gives the number of solutions of $|x - 2| + |x + 1| = k$: a horizontal line $y = k$ misses the graph if $k < 3$, touches the whole flat part if $k = 3$ (infinitely many solutions), and crosses twice if $k > 3$.
Differences of absolute values give "ramps" that level off at both ends.
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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 moduli, $f(x) = |x - a| - |x - b|$ with $a < b$. By observing the vertices and the constant…
The following graph represents a function defined by the difference of two moduli, $f(x) = |x - a| - |x - b|$ with $a < b$. By observing the vertices and the 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 moduli.