Exercises: Intervals

In electronic component design, the voltage tolerance of a sensor is defined by the set of real values $x$ satisfying the following inequality:

$-3 \leq x < 4,5$

Express this set of values using interval notation and state the type of interval (open, closed, or half-open).

Given the following graphical representation on the number line, write the highlighted set in interval notation and in set-builder notation (using inequalities).

A tech company's data server maintains its optimal operating temperature in the interval given by $I = (-10, 35]$. Represent this interval on the number line and determine whether the following temperature values belong to the set:

• $a = -10$

• $b = 0$

• $c = 35$

• $d = 35,1$

Express the result of the following union of sets as a single interval:

$A = (- \infty, 2) \cup [0, 6]$

Use a graphical representation to help you if necessary.

Determine the intersection of the following intervals of real numbers:

$I_1 = [-3, 4]$

$I_2 = (0, + \infty)$

Write the final result both in interval form and as an inequality.

In the study of limits, the concept of a deleted neighbourhood is essential. If we have the neighbourhood $E(3; 0,5)$, which represents all real numbers whose distance to the centre $3$ is less than $0,5$:

1. a) Write this set in interval form.
2. b) Write the deleted neighbourhood $E^*(3; 0,5)$, that is, the same set but excluding the central point.

Match each modulus with its corresponding interval:

1. $|x| \leq 3$

2. $|x| < 5$

3. $|x - 2| \leq 1$

Available intervals:

• $A = ( -5, 5)$

• $B = [ -3, 3]$

• $C = [ 1, 3]$

Write in interval form the domain of a function that only accepts positive values of $x$ less than or equal to $100$, but excluding the exact value $x = 25$ due to a technical singularity. Present the result as a difference of sets or as a union of two intervals.

In an assessment test, Lucía has to find the complement in $\mathbb{R}$ of the storage temperature interval of a drug, given by the set $A = [-8, 2)$. Lucía writes in her exam:

and argues: «The complement consists of flipping the square brackets and parentheses: since $-8$ was closed, I make it open; since $2$ was open, I keep it open; and at infinity I put a square bracket because it includes all the numbers that come after».

1. a) Identify and explain in a reasoned way the two conceptual mistakes Lucía has made in the result (at the endpoint $2$ and in the symbol $+\infty$). Also, comment on whether the endpoint $-8$ in her result is correct and whether the «flipping» rule she uses is valid.

1. b) Write the correct expression of the complement as a union of intervals.

The State Meteorological Agency (AEMET) activates a heat warning in a region of Seville when the maximum temperature $T$ (in $^\circ\text{C}$) is in the interval $I_{\text{AEMET}} = [39, 44)$. Meanwhile, labor regulations in a sector require the suspension of heavy outdoor work if the temperature is at least $41,5\,^\circ\text{C}$, which corresponds to the ray $I_{\text{work}} = [41,5, +\infty)$.

1. a) Determine the temperature interval in which the AEMET warning and the work suspension coincide simultaneously ($I_{\text{AEMET}} \cap I_{\text{work}}$). Justify whether this interval is open, closed, or half-open.
2. b) Express using a ray or union of intervals the temperature range in which at least one of the two measures is active ($I_{\text{AEMET}} \cup I_{\text{work}}$), and reason whether this set is bounded above or bounded below.
3. c) Determine the supremum and the infimum of $I_{\text{AEMET}}$, stating with justification whether they coincide with its maximum or minimum.

In the physics lab, a precision sensor measures the diameter of a metal part. The control system accepts parts whose diameter $x$ (in mm) meets the following tolerance condition expressed by an absolute value:

Represent this situation graphically on the real number line and express the set of accepted values as an interval and as a neighbourhood $E(a; r)$.

You can use this number line, where only the center of the tolerance, $12$, has been marked. Complete it in your notebook: mark the endpoints, shade the accepted values and indicate the radius.

You can do it on paper and send a photo with your answer.

Find the intersection and the union of the following sets of real numbers. Express the final result using interval notation:

$A = \{x \in \mathbb{R} : -2 \leq x < 5\}$

$B = \{x \in \mathbb{R} : |x - 3| \leq 4\}$

To study the continuity of a function at the point $x = 3$, we need to work in a punctured neighborhood of that point with a radius of $0,1$ units.

1. a) Write the definition of that set using the notation $E^*(a, r)$.

1. b) Express it by means of an inequality that includes an absolute value.

1. c) Represent it as the union of two open intervals.

Given the rational function $f(x) = \frac{1}{x^2 - 9}$, determine its domain and express it as a union of intervals. Then, state whether the value $x = 3,0001$ belongs to a neighbourhood with centre $3$ and radius $\delta = 0,001$.

A surveillance drone must maintain a flight altitude $h$ (in metres) such that its distance from the safety altitude of $100$ metres is always less than $5$ metres, but due to turbulence, it cannot fly at exactly $100$ metres.

Express the altitude condition $h$ using:

1. a) A deleted neighbourhood.
2. b) An inequality involving a modulus.
3. c) A union of intervals.

In the formal definition of a limit, when we say that $\lim_{x \to 2} f(x) = 5$, we are stating that for every $\varepsilon>0$ there exists a $\delta>0$ such that, if $0<|x-2|<\delta$, then $|f(x)-5|<\varepsilon$. We take $\varepsilon = 0,2$, which determines a neighborhood of $5$ on the $Y$ axis. The graph shows the function $f(x) = 2x + 1$ and the point of the curve with $x = 2$.

1. a) Determine the interval corresponding to the neighborhood $E(5; 0,2)$.

1. b) Given that the function is $f(x) = 2x + 1$, find the interval of values of $x$ on the $X$ axis whose images fall within the interval obtained in the previous part.

Given the intervals $I = (-\infty, 4]$ and $J = (-2, +\infty)$, perform the following operations and express the result in interval notation and graphically:

1. a) $I \cap J$
2. b) $I \cup J$
3. c) $\mathbb{R} - (I \cap J)$

Determine the set of values of $x$ that satisfy the following quadratic inequality, expressing the result as an interval or union of intervals:

Is this set a neighbourhood of any real number? Justify your answer by finding the centre and the radius if possible.

The goal is to design the algebraic expression of a real function $f(x)$ that models the response of a sensor and combines a square root, a logarithm, and a denominator of the form:

so that its exact domain of definition is the set $D = (3, 4) \cup (4, 9]$.

1. a) Deduce with justification what numerical values the real constants $a$ and $k$ must take.
2. b) Prove step by step that the intersection of the three analytical restrictions (radicand of even index greater than or equal to zero, strictly positive logarithm argument, and non-zero denominator) yields exactly the set $D$.
3. c) State the supremum, infimum, maximum, and minimum of set $D$ (or justify which of them do not exist).

Consider the real function $g(x) = \sqrt{x^2 - 9} + \frac{1}{\ln(7 - x)}$.

1. a) Estimate: before solving formal inequalities, analyze by simple inspection the integer values $x = -4$, $x = 0$, $x = 3$, $x = 6$, and $x = 7$. Justify which of them cannot belong to the domain (pointing out which restriction they violate) and which can. From this test, conjecture whether the ray of very negative values, $(-\infty, -3]$, will be part of the domain.
2. b) Check: formally determine the domain of $g(x)$ by stating and solving the intersection of the three analytical conditions (radicand $\ge 0$, logarithm argument $> 0$, and denominator $\ne 0$). Express the final result in interval notation.
3. c) Assess: compare the exact domain obtained with your initial estimate and explain why the endpoint $3$ is included with a bracket while the endpoint $6$ is excluded with a parenthesis.

Determine the set of real values $x$ that simultaneously satisfy the following rational inequality and the interval defined by the absolute value. Express the final result as an interval or union of intervals:

In the design of a precision part for a telescope, the radius of curvature $R$ must belong to the domain of the following function to ensure structural stability. Find said domain and express it as a union of intervals:

$f(R) = \sqrt{\frac{\ln(R^2 - 8)}{R - 4}}$

A neighbourhood of the point $a = 3$ is defined by the expression $|x - 3| < \delta$. Determine the maximum value of $\delta$ such that all points $x$ in this neighbourhood satisfy that their squares, $x^2$, are at a distance of less than $0,5$ from the value $9$. That is, solve the implication:

Find the set of solutions of the following inequality with nested moduli. Represent the solution on the number line and specify whether it is a bounded set, stating its supremum and infimum if so:

Determine the value of the real parameter $k > 0$ so that the intersection of the sets $A$ and $B$ is an interval of length exactly equal to $3$ units, where:

Hint: express $A$ and $B$ as intervals of the real line and study, depending on the value of $k$, what $A \cap B$ looks like.

A thermal control sensor in a biomass processing plant operates correctly provided that the temperature $T$ (in $^\circ\text{C}$) satisfies the condition $T \in [90, 150]$. If the process temperature changes over time according to the function $T(t) = t^2 - 20t + 175$, where $t$ is the time in minutes ($t \geq 0$), determine the time intervals during which the sensor detects a temperature outside its operating range.

Let sets $S_1$ and $S_2$ be defined by the following conditions. Calculate exactly the set resulting from the operation $(S_1 \cup S_2)^c$ (complement of the union):

Consider the function $g(x) = \log_2\left( |x^2 - 1| - 2 \right)$.

1. a) Determine the set of real values of $x$ for which the function is not defined.

1. b) Express this set as an interval or as a union of intervals.

1. c) Classify each interval obtained as open, closed or half-open.

To calculate the domain of the function $h(x) = \sqrt{\ln\left(\frac{3x - 12}{x + 2}\right)}$, two students propose different approaches:

• Strategy 1 (systematic intersection): solve the condition of the logarithm separately, $\frac{3x - 12}{x + 2} > 0$, obtaining set $S_1$; then solve the condition of the radicand, $\ln\left(\frac{3x - 12}{x + 2}\right) \geq 0 \iff \frac{3x - 12}{x + 2} \geq 1$, obtaining set $S_2$; and finally calculate $S_1 \cap S_2$.

• Strategy 2 (logical implication): argue that if an expression satisfies $\frac{3x - 12}{x + 2} \geq 1$, then it is necessarily greater than zero, so the condition of the logarithm is absorbed by that of the radicand and it is sufficient to solve directly $\frac{3x - 12}{x + 2} \geq 1$, ensuring that $x + 2 \neq 0$.

1. a) Reasonably evaluate whether Strategy 2 is mathematically rigorous or if it carries the risk of omitting solutions or problematic points.
2. b) Apply the strategy you consider most efficient to obtain the domain of $h(x)$ in interval notation.
3. c) Find the complement in $\mathbb{R}$ of the domain obtained and determine whether that complement is a bounded set, stating its supremum and infimum if they exist.

In a chemical plant in Tarragona, the stability of a reaction depends on the pressure $x$ (in bar) through the analytical index:

The safety protocol requires that the plant operate only within the mathematical domain of definition of $S(x)$ and, furthermore, outside the mechanical resonance interval $R = [3,5; 4,8]$.

1. a) Determine the admissible set of pressures (the domain of $S(x)$) using the reasoned intersection of the conditions on the square root, the logarithm argument, and the denominator. Express the result as a union of intervals.
2. b) Determine the safe operating set $C = \text{Dom}(S) \setminus R$. Pay special attention to whether each resulting endpoint is open or closed, and justify why each one is included or excluded.
3. c) Prove whether the set $C$ is bounded above and below. Determine its supremum and infimum with justification, and reason whether either of them reaches the status of maximum or minimum of $C$.