Intervals, Rays, and Neighborhoods

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: $-3 < x \le 5$.

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 $7$.

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

• "Sub-zero temperatures."

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

• "Prices over 1 000 pesos."

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.

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 neighborhood, having previously calculated its center 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.

The price of a video game in an online store fluctuated during the week within the neighborhood $E(490; 36)$ pesos. Determine the maximum and minimum price the video game reached and express it as a closed interval.

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.

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

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.

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

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.

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.

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 $[0; 5.1)$.

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 $(I \cap J) \cup K$.

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.

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 neighborhood with center $0.8\text{ UA}$ 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.