Exercises: Intervals, rays and neighbourhoods

Look at the graphical representation of the following set of real numbers and express it using interval notation and as an inequality.

Express the following inequality as an interval and represent it approximately on a number line: $-3 < x \le 5$.

Write the open interval $(2, 10)$ as a neighbourhood $E(a, r)$.

Express the neighbourhood $E(-1, 4)$ as an interval and convert it into an inequality.

State which of the following numbers belong to the interval $[-1,5; 3)$:

$-2$, $-1,5$, $0$, $2,99$, $3$, $\pi$

Write as a half-line the set of all real numbers that are less than or equal to $7$.

Express as an inequality and as an interval the set of real numbers whose distance from zero is strictly less than $5$.

Match each of the following statements with its corresponding mathematical interval:

• "Sub-zero temperatures."

• "Ages between 12 and 16 years, inclusive."

• "Prices over 50 euros."

In an exam, Marcos must express the set of real numbers satisfying $|x - 2| > 4$ as a union of intervals. Marcos writes:

"Since the centre is $2$ and the radius is $4$, I calculate $2 - 4 = -2$ and $2 + 4 = 6$. Therefore, the solution is the open interval $(-2, 6)$".

1. a) Explain what conceptual mistake Marcos makes when interpreting the absolute value inequality in terms of distance.
2. b) Correct his solution and write the correct solution set as the union of two rays in interval notation.

A set of real numbers is given by the union of rays $(-\infty, -1) \cup (7, +\infty)$.

1. a) Calculate the midpoint (centre) and the distance from that centre to the endpoints $-1$ and $7$ (radius).
2. b) Write an absolute value inequality of the form $|x - a| > r$ whose solution matches this set exactly.
3. c) Which inequality sign should you change, and to what, so that the solution set becomes the closed interval $[-1, 7]$? Justify your answer.

In nutrition, Body Mass Index (BMI) is considered healthy if it lies within the interval $[18,5; 24,9]$. Express this range as a closed neighbourhood, having previously calculated its centre and radius.

A machine produces steel shafts with an ideal length of $120\text{ mm}$. Quality control permits a maximum error of $0,5\text{ mm}$. Express the range of permitted lengths as a neighbourhood, as an inequality with modulus, and finally as an interval.

Given the intervals $A = [-4, 3)$ and $B = (0, 6]$, determine the interval resulting from the union $A \cup B$ and the intersection $A \cap B$.

In an aquarium, clownfish thrive at temperatures in the interval $(24, 28]$ ºC, whereas angelfish require temperatures in the interval $[26, 30)$ ºC. Determine the temperature interval in which both species can coexist and express it as an inequality.

Find all the integers that belong simultaneously to the intervals $(-3,2; 4]$ and $[0; 5,1)$.

Express using modulus the set of real numbers whose distance from the number $-2$ is less than or equal to $6$, and write the resulting interval.

The price of a video game in an online shop fluctuated during the week within the neighbourhood $E(24,5; 1,8)$ euros. Determine the maximum and minimum price the video game reached and express it as a closed interval.

Solve the modulus inequality $|x - 4| < 3$ and express the solution as an interval and as a neighbourhood.

A pharmaceutical laboratory packages saline solution in vials. The nominal volume must be $100\text{ ml}$, and internal quality control considers acceptable any vial whose volume $V$ satisfies $|V - 100| \le 1,5$. Moreover, the calibration machine on the distribution line automatically rejects any vial that does not belong to the interval $[99; 101,2]$.

1. a) Before calculating the exact numerical endpoints, give a reasoned estimate of whether the distribution machine will accept all vials approved by the laboratory or if it will discard some. Justify your prediction using the centres and tolerance margins of both criteria.
2. b) Express the laboratory's criterion as an interval and calculate the intersection of both sets, indicating what this result represents in the context of the problem.
3. c) Check whether your initial estimate was correct and determine the range of volumes that, while valid for the laboratory, are rejected by the distribution machine.

The State Meteorological Agency (AEMET) establishes the following heat warning levels in a region according to the forecast maximum temperature $T$ (in $^{\circ}\text{C}$):

• Green level (no risk): $T < 36$

• Yellow level (low risk): $36 \le T < 39$

• Orange level (significant risk): $39 \le T < 42$

• Red level (extreme risk): $T \ge 42$

For the first days of August, meteorological models indicate that the maximum temperature will satisfy the condition $|T - 38,5| \le 2,5$.

1. a) Translate the weather forecast into a single closed interval and also express it as a double inequality.
2. b) Determine which AEMET warning levels will be activated according to this forecast and write the subinterval of temperatures associated with each activated level.
3. c) Municipal bylaws prohibit roadworks on public roads if the orange or red level is activated. For which temperature interval of the forecast will roadworks be prohibited?

A bridge is designed to support loads $C$ (in kg) satisfying the condition $|C - 4500| \le 1500$. Furthermore, a local regulation prohibits the passage of vehicles exceeding $5500\text{ kg}$. Determine the actual interval of loads that can travel across the bridge while meeting both conditions.

Given the sets $I = (-\infty, 2]$, $J = (-1, 5)$ and $K = [3, 8]$, calculate the interval resulting from the operation $(I \cap J) \cup K$.

Express the closed interval $[3,2; 8,8]$ as a neighbourhood $E(a, r)$ and then convert it into a modulus inequality of the form $|x - a| \le r$. Briefly explain what the parameters $a$ and $r$ represent geometrically on the number line.

For swimming pool water to be safe, its pH must be strictly in the interval $[7,2; 7,6]$. If we have an electronic meter with a maximum absolute error of $0,1$, determine the interval in which the device reading must fall to be absolutely certain that the actual pH is safe. Express it as a neighbourhood as well.

Find the interval of real numbers $x$ that simultaneously satisfy the inequalities $3x - 2 < 7$ and $5 - x < 4$. Once found, express it as a neighbourhood.

A screw manufacturer accepts screws whose length $L$ (in mm) satisfies the condition $|L - 30| < 0,5$. A batch of screws is received whose lengths vary uniformly in the interval $[29,2; 30,8]$. What percentage of the length range of this batch will be rejected by quality control?

The "habitable zone" of a red dwarf star is defined by a neighbourhood with centre $0,8\text{ UA}$ and radius $0,2\text{ UA}$. An exoplanet has an elliptical orbit whose distance to the star varies within the interval $[0,5; 1,1]\text{ UA}$. Determine which intervals of the planet's orbit fall outside the habitable zone.

Determine the value of the parameter $k$ such that the solution interval of the inequality $|x - k| \le 5$ has upper endpoint $12$. Once $k$ has been calculated, state the lower endpoint of the interval and express it as a neighbourhood.

To solve the absolute value inequality $|x - 4| \le 3$ and express it as an interval, two students follow different strategies:

• Approach 1 (Geometric): Andrea interprets $|x - 4| \le 3$ as "the numbers whose distance to $4$ is less than or equal to $3$". She identifies the centre $c = 4$ and the radius $r = 3$, obtaining $[4 - 3, 4 + 3] = [1, 7]$.

• Approach 2 (Algebraic): Berta sets up the double inequality $-3 \le x - 4 \le 3$ and adds $4$ to all three parts to isolate $x$, obtaining $1 \le x \le 7$, that is, $[1, 7]$.

1. a) Compare both approaches by applying them to the inequality $|x + 2| < 5$. Explain how each works and verify that both lead to the same interval.
2. b) Now solve the inequality $|2x - 5| \le 3$ using both methods. In Approach 1, indicate what prior algebraic transformation you must perform to be able to identify the centre and the radius on the real line.
3. c) In light of the previous section, analyse the advantages and disadvantages of each method when the coefficient of the unknown $x$ is not $1$. Justify which of the two approaches is more convenient and reliable in that case.

Consider the set $A = \{x \in \mathbb{R} \mid |x - 3| < 5\}$ and the closed interval $B = [p, q]$, where $p$ and $q$ are real numbers with $p < q$.

1. a) Express set $A$ as an open interval and find its midpoint (centre) and its length.
2. b) Determine with reasoning the exact values of $p$ and $q$ so that these two conditions are met simultaneously:

1. The intersection $A \cap B$ is exactly the half-open interval $[1, 8)$.

2. The length of interval $B$ is equal to $10$.

1. c) With the values found for $p$ and $q$, calculate the union $A \cup B$ and justify why the resulting set is a single connected interval and not two disjoint intervals.

Unit Challenge. Have you ever wondered what mathematics underpins the objects you use every day to study? Investigate the numerical nature of your screens and materials.

1. Measurement: Measure with a ruler the length and width (in cm) of a DIN-A4 sheet of paper and the screen of your mobile phone or computer.

2. Numerical analysis: Open a spreadsheet and calculate the length/width ratio of both objects. Research the standard theoretical values: the paper's ISO 216 proportion ($\sqrt{2}$) and your screen's standard (usually $16/9$). Classify them with justification into rational or irrational based on their decimal expansion.

3. Precision check: Look up the manufacturer's official measurements for your screen and those of the A4 standard ($29,7 \times 21,0\text{ cm}$). In the spreadsheet, calculate the absolute error and the percentage relative error made in your measurements, bounding and rounding the results to two decimal places with a comma ($0,01\%$).

4. Final report: Present a comparative table and a brief reflection: why do you think paper is designed with an irrational number and video formats with a rational fraction?

Evaluation criteria:

• Correct and reasoned mathematical classification of rational and irrational numbers.

• Accuracy in calculating approximations, absolute error, and relative error in the spreadsheet.

• Quality of the report, rigor in using the decimal comma, and consistency in the conclusions.