Exercises: The real line

Express as an interval and as an inequality the set of real numbers that are greater than or equal to $-3$ and strictly less than $5$.

Write as an interval the set of real numbers that meet the condition $|x| \le 7$.

Given the intervals $A = [-2, 4)$ and $B = (1, 6]$, determine and express as a single interval the result of the union $A \cup B$ and the intersection $A \cap B$.

Express the neighborhood $E(-1, 3)$ as a numerical interval and describe it using a modulus inequality.

Calculate the distance on the real line between the points $A = -4,5$ and $B = 2,3$, and express the mathematical operation using modulus notation.

State which of the following real numbers belong to the interval $[-1, \pi)$:

$-1,01$, $-1$, $3,14$, $\frac{22}{7}$, $\sqrt{10}$.

Find the values of $x$ that satisfy the equation $|x - 4| = 9$.

Given the open interval $(2, 8)$, express it as a neighborhood, explicitly stating its center $a$ and its radius $r$.

In a biology laboratory, an incubator must maintain a temperature $T$ in degrees Fahrenheit that does not differ from $37^{\circ}\text{C}$ by more than $0,5^{\circ}\text{C}$. Express this tolerance condition using a modulus inequality and determine the interval of permitted temperatures.

A cylindrical engine part has an ideal diameter of $50\text{ mm}$. Quality control rejects the part if the measured diameter $d$ satisfies $|d - 50| > 0,02$. Write the interval of diameters that are accepted by quality control.

Explain the steps to represent the irrational number $\sqrt{13}$ exactly on the real line using Pythagoras' theorem.

Given the sets $A = (-\infty, 2]$, $B = (-1, 5)$ and $C = [0, +\infty)$, calculate $(A \cap B) \cup C$ and express the result as a single numerical interval.

Solve the inequality $|2x - 5| \le 3$ and represent the solution set as an interval.

The price of a stock market share fluctuates around a central value. If the fluctuation interval during the week was $[112,5 , 127,5]$, express this range using the neighborhood notation $E(a, r)$ and a modulus inequality of the form $|x - a| \le r$.

Consider the interval $I = [-3, 4)$. Determine the complement set $\mathbb{R} \setminus I$ and express it as the union of two intervals.

In an archery competition, a shot is considered "excellent" if the arrow hits at a distance $d$ from the center of the target that satisfies $|d| < 2,5\text{ cm}$. If the target is positioned so that its center is the origin of coordinates on a one-dimensional axis, state the interval of "excellent" positions.

Determine the values of the parameter $k \in \mathbb{R}$ for which the intersection of the intervals $A = (-\infty, 2k]$ and $B = (k + 3, +\infty)$ is an interval of length exactly equal to $5$.

Find the set of real numbers $x$ that satisfy the inequality $|x + 2| + |x - 3| \le 7$. Express the result as an interval.

In the design of a building's facade, the golden ratio $\phi = \frac{1 + \sqrt{5}}{2}$ is used. Based on the attached figure, justify analytically why the construction shown determines exactly the value of $\phi$ on the real line.

Express as a union of intervals the set of all real numbers that satisfy the quadratic inequality $x^2 - 4x - 5 > 0$.

Prove, by finding an explicit example, that between the rational numbers $\frac{10}{7}$ and $\frac{143}{100}$ there is at least one irrational number.

Solve the inequality with nested moduli $||x - 1| - 2| \le 1$ and express the final solution as the union of two closed intervals.

In an optics experiment, the position $x$ of a lens along a one-dimensional rail must simultaneously satisfy two conditions for the image to be in focus: the distance to the light source $A$ (located at $x=10$) must be greater than $5\text{ cm}$, and the distance to the screen $B$ (located at $x=30$) must be less than $12\text{ cm}$. Express these conditions using a system of modulus inequalities and determine the valid interval for the position $x$.

The measurement of the radius of a sphere is $r = 5,0 \pm 0,1\text{ cm}$. Using the properties of inequalities and intervals on the real line, determine the strict interval in which the exact volume of the sphere, $V = \frac{4}{3}\pi r^3$, is found. Consider the unapproximated value of $\pi$ at the endpoints of the resulting interval.