Question

How do you solve and write the following in interval notation: $t$ is at least $10$ and at most $22$ ?

Answer 1

These types of problems are very straight forward and are easy to solve and represent. For these types of problems, we first need to understand what at least and at most means for representing a number. In such problems we are given a real number in between a certain range, and we need to represent that in the form of brackets, whether it would be an open bracket or a close bracket.

Complete step-by-step solution:
Now, starting off with the solution of the problem, we first assume a real number, say $x$. Now, according to the problem, we see it’s given that the value of $x$is at least $10$. Now, this statement means that the minimum possible value of $x$ is $10$ . Thus we can write,
$x\ge 10$
We have another statement saying that the value of $x$ is at most $22$ . From this we conclude that the maximum possible value of $x$ can be,
$x\le 22$
Now, combining both of these statements we can safely say that $x$ is such a real number which is greater than or equal to $10$ and is less than or equal to $22$ . In other words we can say that the value of $x$ lies in between a $10$ and $22$. When we have a value equal to that of another value in terms of range, we use a closed bracket. When we have a value less than or greater than another value, we use an open bracket. Here since both the end points are equal, i.e. $x\ge 10$ and $x\le 22$ , we use closed brackets.
Finally we can write,
$x\in \left[ 10,22 \right]$

Note: For such types of problems, we first need to read the problem statement very carefully, as what is required to find. We also have to keep a close eye about the equality sign of the end points. If there is an equality sign at the end points, we need to consider a closed bracket, and if nothing has been mentioned, we need to use the open bracket. This range of numbers represent all the real valued numbers between the two end points.