Intervals, rays and neighbourhoods
A range of safe temperatures, a healthy BMI, the tolerance margin of a mechanical part: all of them are pieces of the real line. To work with them precisely you need formal notation: intervals, rays and neighbourhoods.
Intervals and rays
Rays are intervals with only one endpoint; the other one "goes off" to infinity, which is never included because it is not a number.
Look at the graphical representation of the following set of real numbers and express it using interval notation and as an inequality.
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: $-3 < x \le 5$.
Express the following inequality as an interval and represent it approximately on a number line: $-3 < x \le 5$.
Write as a half-line the set of all real numbers that are less than or equal to $7$.
Write as a half-line the set of all real numbers that are less than or equal to $7$.
Neighbourhoods
A neighbourhood describes an interval in terms of its centre and how far it extends to each side: it is the natural way of talking about "tolerance" or "margin of error".
Write the open interval $(2, 10)$ as a neighbourhood $E(a, r)$.
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.
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 neighbourhood, having previously…
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.
Absolute value, distance and inequalities
The absolute value $|x - a|$ measures the distance between $x$ and $a$ on the real line, regardless of which of the two is larger. This connects directly with neighbourhoods and lets you solve inequalities systematically.
Express as an inequality and as an interval the set of real numbers whose distance from zero is strictly less than $5$.
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.
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.
Solve the modulus inequality $|x - 4| < 3$ and express the solution as an interval and as a neighbourhood.
Union and intersection of intervals
Intervals are sets, so they can be combined using set operations: the union ($\cup$) puts together everything that is in either of the two; the intersection ($\cap$) keeps only what is common to both.
Given the intervals $A = [-4, 3)$ and $B = (0, 6]$, determine the interval resulting from the union $A \cup B$ and the intersection $A \cap B$.
Given the intervals $A = [-4, 3)$ and $B = (0, 6]$, determine the interval resulting from the union $A \cup B$ and the intersection $A \cap B$.
Find all the integers that belong simultaneously to the intervals $(-3,2; 4]$ and $[0; 5,1)$.
Find all the integers that belong simultaneously to the intervals $(-3,2; 4]$ and $[0; 5,1)$.
Given the sets $I = (-\infty, 2]$, $J = (-1, 5)$ and $K = [3, 8]$, calculate the interval resulting from the operation $(I \cap J) \cup K$.
Given the sets $I = (-\infty, 2]$, $J = (-1, 5)$ and $K = [3, 8]$, calculate the interval resulting from the operation $(I \cap J) \cup K$.