Exercises: The number 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 satisfying 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 of the intersection $A \cap B$.

Express the neighbourhood $E(-1, 3)$ as a numerical interval and describe it using an absolute value inequality.

Calculate the distance on the number line between points $A = -4.5$ and $B = 2.3$, and express the mathematical operation using absolute value 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 in neighborhood notation, explicitly indicating its center $a$ and its radius $r$.

In a biology laboratory, an incubator must maintain a temperature $T$ in degrees Celsius that does not differ from $37^{\circ}\text{C}$ by more than $0.5^{\circ}\text{C}$. Express this tolerance condition using an absolute value 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 number 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 on the stock market fluctuates around a central value. If the fluctuation interval during the week was $[112.5 , 127.5]$, express this range using neighbourhood notation $E(a, r)$ and using an absolute value inequality of the form $|x - a| \le r$.

Consider the interval $I = [-3, 4)$. Determine the complementary 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 such 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.

The figure shows the graph of the function $f(x) = |x + 2| + |x - 3|$ and the line $y = 7$, so that you can check your result.

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

Express as a union of intervals the set of all real numbers satisfying 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 absolute values $||x - 1| - 2| \le 1$ 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 absolute value 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 number line, determine the strict interval in which the exact volume of the sphere, $V = \frac{4}{3}\pi r^3$, lies. Consider the value of $\pi$ unapproximated at the endpoints of the resulting interval.