Exercises: Absolute value of a real number

Calculate the numerical value of the following expression: $|-7| - |4 - 9| + |-3|$.

Solve the absolute value equation: $|3x| = 12$.

Express the inequality $|x| \le 5$ as an interval.

Write the interval $(-8, 8)$ using absolute value notation.

Calculate the distance on the number line between points $A = -6$ and $B = 11$ using the absolute value.

Solve the equation $|x - 4| = 3$.

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.

Lucas tries to solve the equation $|3x - 6| = -9$ on the board and writes the following steps:

$3x - 6 = -9 \Rightarrow 3x = -3 \Rightarrow x = -1$

$3x - 6 = 9 \Rightarrow 3x = 15 \Rightarrow x = 5$

Therefore, he confidently states that the equation has two solutions: $x = -1$ and $x = 5$.

1. a) Without needing to substitute the obtained values into the original equation, explain what conceptual error Lucas makes right from the first step.
2. b) Determine the true solution set and justify your answer based on the geometric interpretation of the absolute value as a distance.
3. c) What condition should the number on the other side of the equality satisfy for the method of splitting into two cases with opposite signs to have been valid?

In a metrology laboratory, industrial sensors are tested according to the permissible tolerance margin in their readings.

1. a) Write an absolute value inequality of the form $|x - c| \le r$ whose solution set coincides exactly with the closed interval $[-3, 7]$. Deduce with reasoning the values of the center $c$ and the tolerance radius $r$ from the endpoints of the interval.
2. b) Without needing to recalculate the center or the radius, modify the previous inequality to describe the sensors that must be discarded for exceeding the permitted tolerance, that is, those whose reading belongs to the set $(-\infty, -3) \cup (7, +\infty)$.

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.

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 centre $c$ and the radius $r$.

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.

Find all real values of $x$ that satisfy the equation with two absolute values: $|3x - 1| = |x + 5|$.

The price of a stock on the stock market today is $60$ euros. 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 envisaged by the prediction?

Simplify the algebraic expression $|x - 3| + |x + 2|$ knowing that the variable $x$ belongs to the interval $(-2, 3)$.

Find all real numbers $x$ such that their distance to point $7$ is exactly double their distance to point $-2$.

The State Meteorological Agency (AEMET) issues a yellow weather warning for heat in the countryside of Cordoba when the forecast maximum temperature $T$ (in $^\circ\text{C}$) satisfies the inequality $|T - 39,5| \le 2,5$.

1. a) Solve the inequality and interpret the result by determining the closed temperature interval that this warning level covers. Clearly indicate the central reference temperature and the maximum permitted oscillation.
2. b) AEMET elevates the situation to an orange warning if the temperature strictly exceeds the upper limit of the yellow warning but does not reach $44,0\,^\circ\text{C}$. Model this new temperature interval using a single absolute value inequality of the form $|T - c| < r$.
3. c) If an automatic weather station records $36,4\,^\circ\text{C}$, justify mathematically whether or not it is within the yellow warning range by calculating the numerical value of $|T - 39,5|$ and interpreting the resulting distance.

Consider the equation with two absolute values $|x - 1| + |x - 7| = 10$.

1. a) Geometrically interpret $|x - 1|$ and $|x - 7|$ as the distances from a real number $x$ to the points $1$ and $7$ on the number line. What is the sum of these two distances if $x$ lies at any point inside the segment $[1, 7]$?
2. b) From the previous reasoning, explain why it is impossible for solutions to exist within the interval $[1, 7]$ and estimate with reasoning, taking advantage of the symmetry of the real line and without performing algebraic calculations, what the two exact solutions of the equation must be.
3. c) Solve the equation formally by studying the three intervals determined by the critical points $x = 1$ and $x = 7$ (or by defining the left-hand side as a piecewise function) and check whether your initial estimation was correct.

Solve the quadratic inequality with absolute value: $|x^2 - 2x| < 3$.

The navigation system of a submarine requires its ideal depth $h$ (in metres, 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.

Discuss the number of solutions of the equation $|x - 4| + |x + 1| = k$ depending on the values of the real parameter $k$.

Solve the following equation containing nested absolute values: $||x - 2| - 5| = 3$.

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.

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.

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 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 $y - t$.

To solve the inequality $|2x - 3| \le |x + 3|$, two Bachillerato students propose different approaches:

• Method A: Divide the real line into three intervals using the critical points where each absolute value vanishes ($x = -3$ and $x = 1,5$), rewrite the expression without bars in each region, and solve the three resulting systems.

• Method B: Since both sides are non-negative for any $x \in \mathbb{R}$, square both sides to obtain $(2x - 3)^2 \le (x + 3)^2$, and solve by factoring a difference of squares $(2x - 3)^2 - (x + 3)^2 \le 0$.

1. a) Apply Method B by factoring the difference of squares (without expanding the squared binomials) to find the solution set in the form of an interval.
2. b) Formally justify why Method B is mathematically rigorous in this exercise and explain why it would cease to be directly applicable if the inequality were $|2x - 3| \le |x + 3| - 4$.
3. c) Compare the algebraic workload and the risk of error between both methods. Which of the two is more advantageous for inequalities of the form $|A(x)| \le |B(x)|$?

In a written test, Mateo attempts to deduce and apply absolute value properties, but makes a chain of conceptual errors:

"By the triangle inequality we know that $|a| = |(a - b) + b| \le |a - b| + |b|$, from which subtracting $|b|$ yields $|a| - |b| \le |a - b|$. Since distance is never negative, we can conclude that always $|a - b| = |a| - |b|$. Therefore, to solve the inequality $|x - 4| \ge 5$, it suffices to write $|x| - 4 \ge 5$, that is, $|x| \ge 9$, whose solution is $(-\infty, -9] \cup [9, +\infty)$".

1. a) Point out precisely the two flaws in logical reasoning made by Mateo and provide a simple numerical counterexample with specific values of $a$ and $b$ that refutes the equality $|a - b| = |a| - |b|$.
2. b) From Mateo's correct step ($|a| - |b| \le |a - b|$), swap the roles of $a$ and $b$ to rigorously deduce the reverse triangle inequality: $\big||a| - |b|\big| \le |a - b|$. What condition must the signs of $a$ and $b$ satisfy for this expression to achieve equality?
3. c) Correctly solve the inequality $|x - 4| \ge 5$ and interpret on the number line why the solution obtained by Mateo was shifted with respect to the true solution.