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 neighbourhood notation, explicitly indicating its centre $a$ and its radius $r$.

Marcos is working on an assignment about the real line and has to calculate the complement of the interval $I = [-3, 5)$ in $\mathbb{R}$. In his notebook, he writes word for word:

"The complement is the opposite of the interval, so I change the brackets and the signs: $\mathbb{R} \setminus I = (-\infty, -3] \cup [5, +\infty)$".

1. a) Explain in detail why Marcos' reasoning is flawed, pointing out what specifically happens with the numbers $-3$ and $5$ (indicate which set each of them belongs to and why they should or should not be in the complement).
2. b) Write the correct solution for $\mathbb{R} \setminus I$ both as a union of intervals and as a logical inequality using the connective "or".

The State Meteorological Agency (AEMET) issues a yellow alert for severe frost in a region of the province of Teruel when the forecast minimum temperature, $T$ (in $^{\circ}\text{C}$), is within a distance of no more than $3\,^{\circ}\text{C}$ from a reference value of $-5\,^{\circ}\text{C}$.

1. a) Mathematically model this condition by writing an absolute value inequality of the form $|T - a| \le r$, identifying the centre $a$ and the tolerance radius $r$.
2. b) Express the set of temperatures under alert as a closed interval and as a double inequality of the form $c \le T \le d$.
3. c) If $-8,5\,^{\circ}\text{C}$ is recorded at one weather station and $-1,8\,^{\circ}\text{C}$ at another, determine with reasoning whether the alert condition was met at either of them by calculating the distance $d(T, -5)$ in each case.

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 centre of the target that satisfies $|d| < 2,5\text{ cm}$. If the target is positioned such that its centre is the origin of coordinates on a one-dimensional axis, state the interval of "excellent" positions.

Consider the set of real numbers satisfying the quadratic inequality $x^2 - 4x - 5 < 0$.

1. a) Estimate: Without using the quadratic formula, observe that the vertex of the associated parabola is located at $x = 2$ and that for this value the trinomial is negative ($2^2 - 4 \cdot 2 - 5 = -9$). Knowing that for $x = 0$ it takes the value $-5$ and for $x = 6$ it takes the positive value $7$, give a reasoned estimate of between which two integers the interval of solutions must lie and what its midpoint should be.
2. b) Calculate: Formally solve the inequality by finding the roots of the quadratic equation and deducing the sign of the parabola on each interval of the real line.
3. c) Check and formalise: Express the exact solution as an open interval and as a neighbourhood $E(a, r)$, identifying its centre $a$ and its radius $r$. Check whether your previous estimate was consistent and reflect on how the symmetry of the parabola with respect to its vertex determines the centre of the neighbourhood.

A bounded subset $S \subset \mathbb{R}$ satisfies two geometric conditions on the real line:

1. Its complement is the union of two disjoint rays: $\mathbb{R} \setminus S = (-\infty, -2] \cup [8, +\infty)$.

2. All of its points are at a distance of less than $r$ from a central point $c$.

1. a) Reconstruct the set $S$ by writing it as a bounded interval. Determine with reasoning the centre $c$ and the radius $r$ to express $S$ in neighbourhood notation $E(c, r)$ and as an absolute value inequality.
2. b) A classmate claims: "We can directly express the complement $\mathbb{R} \setminus S$ using an inequality of the form $x^2 + bx + d \le 0$ by suitably choosing the values of $b$ and $d$". Justify why your classmate's claim is false by analysing the sign of any parabola with a positive quadratic coefficient ($a > 0$), and write the correct quadratic inequality (with leading coefficient $1$) whose solution set is exactly $\mathbb{R} \setminus S$.

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.

To solve the inequality $|x + 3| + |x - 5| \le 12$, two students propose different approaches:

• Method A (Geometric, proposed by Lucía): Interpret the expression as the sum of distances on the real line from a point $x$ to the points $A = -3$ and $B = 5$, that is, $d(x, -3) + d(x, 5) \le 12$. Reason what the sum of distances equals if $x$ is inside the segment $[-3, 5]$, and deduce how far $x$ can move outwards before exceeding the limit of $12$.

• Method B (Piecewise analytical, proposed by Darío): Split the real line into three intervals bounded by $x = -3$ and $x = 5$, remove the absolute value bars according to the sign of each binomial in each region, and solve the resulting inequalities.

1. a) Solve the inequality using Method A, detailing the geometric reasoning on the line.
2. b) Solve the inequality using Method B, showing the analytical breakdown for all three cases.
3. c) Compare both procedures: which one is more efficient in this particular case and why? How would the applicability of the geometric method change if the inequality were $|2x + 3| + |x - 5| \le 12$?

Consider the closed interval depending on a real parameter $k$, $A_k = [k - 1, 2k + 3]$, and the set $B = \{x \in \mathbb{R} : |x - 2| < 5\}$.

1. a) Characterise the set $B$ as an open interval and calculate its complement $\mathbb{R} \setminus B$, expressing it as the union of two disjoint rays and indicating with precision whether the endpoints are included.
2. b) Determine the algebraic condition on $k$ for $A_k$ to be a non-degenerate interval (containing more than one point) and find the expression for the length of the interval $A_k$ as a function of $k$.
3. c) We want to adjust the parameter $k$ such that the intersection between the interval $A_k$ and the complement of $B$ consists of a single real number, that is, $A_k \cap (\mathbb{R} \setminus B) = \{p\}$. Deduce with reasoning all possible values of $k$ that satisfy this condition and determine the value of the contact point $p$ in each case.