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:

• $a = -10$

• $b = 0$

• $c = 35$

• $d = 35.1$

Match each modulus with its corresponding interval:

1. $|x| \leq 3$

2. $|x| < 5$

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

Available intervals:

• $A = ( -5, 5)$

• $B = [ -3, 3]$

• $C = [ 1, 3]$

In the study of limits, the concept of a deleted neighborhood is essential. If we have the neighborhood $E(3; 0.5)$, 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 neighborhood $E^*(3; 0.5)$, that is, the same set but excluding the central point.

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.

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.

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 an 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.

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:

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.

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

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

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

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.

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) $\mathbb{R} - (I \cap J)$

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):

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:

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.

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 $\delta = 0.001$.

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:

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

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.

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.