Intervals, Inequalities, and Neighborhoods

In the design of electronic components, the voltage tolerance of a sensor is defined by the set of real values $x$ that satisfy the following inequality condition:

$-3 \leq x < 4,5$

Express this set of values using interval notation and indicate what type of interval it is (open, closed or half-open).

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

A data server of a technology company maintains its optimum operating temperature in the interval given by $I = (-10, 35]$. Represent this interval on the real 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 physics laboratory, a precision sensor measures the diameter of a metal part. The control system accepts parts whose diameter $x$ (in inches) satisfies the following tolerance condition expressed using a modulus:

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

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

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

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

In the study of limits, the concept of a deleted neighborhood is fundamental. 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.

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

1. a) Write the definition of this set using the notation $E^*(a, r)$.
2. b) Express it using an inequality that includes a modulus.
3. c) Represent it as the union of two open intervals.

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

Express the condition for the height $h$ using:

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

Find 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 calculating the center and the radius if possible.

A thermal control sensor in a biomass processing plant operates correctly as long as the temperature $T$ (in $^\circ\text{C}$) satisfies the condition $T \in [90, 150]$. If the process temperature evolves 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.

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

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 you consider it necessary.

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 the 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):

Write in interval form the domain of definition 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. Express the result as a difference of sets or as a union of two intervals.

Given the rational function $f(x) = \frac{1}{x^2 - 9}$, determine its domain of definition and express it as a union of intervals. Subsequently, indicate if 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 this domain and express it as a union of open intervals:

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

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:

In the formal definition of a limit, when we say that $\lim_{x \to 2} f(x) = 5$, we state 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.

1. a) Determine the interval corresponding to the neighborhood $E(5; 0,2)$.
2. b) If we know 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 section.

A neighborhood of the point $a = 3$ is defined by the expression $|x - 3| < \delta$. Determine the maximum value of $\delta$ so that all points $x$ in this neighborhood satisfy that their squares, $x^2$, are at a distance of less than $0.5$ from the value $9$. That is, solve the implication: