Exercises: Intervals, half-lines and neighbourhoods

Observe 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 neighborhood $E(a, r)$.

Express the neighborhood $E(-1, 4)$ as an interval and translate 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 ray 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 phrases with its corresponding mathematical interval:

• "Temperatures below zero."

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

• "Prices greater than 50 dollars."

In the field of nutrition, the Body Mass Index (BMI) is considered healthy if it is in the interval $[18.5, 24.9]$. Express this range as a closed neighborhood, calculating its center and radius first.

A machine produces steel shafts whose ideal length is $120\text{ mm}$. Quality control allows a maximum error of $0.5\text{ mm}$. Express the range of permitted lengths as a neighborhood, as an inequality with a modulus, and finally as an interval.

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

In a laboratory, the Alpha strain of bacteria survives at temperatures in the interval $(-5, 20]$ ºF, while the Beta strain survives in the interval $[10, 35)$ ºF. Determine the temperature interval in which both strains 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 a 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 share price of a technology company fluctuated during the week within the neighborhood $E(24.5, 1.8)$ dollars. Determine the maximum and minimum price reached by the shares and express it as a closed interval.

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

A bridge is designed to support loads $C$ (in pounds) that satisfy the condition $|C - 4500| \le 1500$. On the other hand, a local regulation prohibits the passage of vehicles exceeding $5500\text{ lb}$. Determine the real interval of loads that can cross the bridge satisfying both conditions.

Let the sets be $I = (-\infty, 2]$, $J = (-1, 5)$ and $K = [3, 8]$. Calculate the resulting interval of the operation $(I \cap J) \cup K$.

Express the closed interval $[3.2, 8.8]$ as a neighborhood $E(a, r)$ and then translate it into an inequality with a modulus of the form $|x - a| \le r$. Briefly explain what the parameters $a$ and $r$ represent geometrically on the real line.

For the water in a swimming pool to be safe, its pH must be strictly in the interval $[7.2, 7.6]$. If we have an electronic meter that has a maximum absolute error of $0.1$, determine in which interval the device must give the reading so that we have absolute certainty that the real pH is safe. Express it also as a neighborhood.

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

A screw manufacturer accepts those whose length $L$ (in inches) 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 from the star varies in 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 $k$ so that the solution interval of the inequality $|x - k| \le 5$ has the number $12$ as its upper bound. Once $k$ is calculated, state what the lower bound of the interval is and express it as a neighborhood.