Exercises: Modulus of a real number
Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.
Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.
Express the inequality $|x| \le 5$ as an interval.
Express the inequality $|x| \le 5$ as an interval.
Write the interval $(-8, 8)$ using modulus notation.
Write the interval $(-8, 8)$ using modulus notation.
Calculate the distance on the real line between the points $A = -6$ and $B = 11$ using the modulus.
Calculate the distance on the real line between the points $A = -6$ and $B = 11$ using the modulus.
Express the inequality $|x + 2| > 4$ as a union of intervals.
Express the inequality $|x + 2| > 4$ as a union of intervals.
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.
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.
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.
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 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.
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: $|3x - 1| = |x + 5|$.
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?
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 $(-2, 3)$.
Find all the real numbers $x$ such that their distance to the point $7$ is exactly twice their distance to the point $-2$.
Find all the real numbers $x$ such that their distance to the point $7$ is exactly twice their distance to the point $-2$.
Solve the quadratic modulus inequality: $|x^2 - 2x| < 3$.
Solve the quadratic modulus inequality: $|x^2 - 2x| < 3$.
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.
Discuss the number of solutions to the equation $|x - 4| + |x + 1| = k$ depending on the values of the real parameter $k$.
Discuss the number of solutions to the equation $|x - 4| + |x + 1| = k$ depending on the values of the real parameter $k$.
Solve the following equation containing nested moduli: $||x - 2| - 5| = 3$.
Solve the following equation containing nested moduli: $||x - 2| - 5| = 3$.
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 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.
Solve the fractional inequality $\frac{3}{|x - 1|} \ge 2$.
Solve the fractional inequality $\frac{3}{|x - 1|} \ge 2$.
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 $y - t$.