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 center $3$ 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 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 neighborhood with center $3$ and radius $\delta = 0.001$.

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

Express the altitude condition $h$ using:

1. a) A deleted neighborhood.
2. b) An inequality involving absolute value.
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 neighborhood of any real number? Justify your answer by finding the center and the radius if possible.

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.