Question

How do you write x > -17 as a set notation and interval notation? \[\]

Answer 1

Hint: We recall how to represent a collection of numbers in the set builder notation using a predicate. We also recall how to write a collection number within two numbers using intervals. We use $x> -17$ as a predicate to write in set builder notation and find upper limits and lower limits to write in interval form.

Complete step-by-step solution:
We know that we can represent a collection of numbers in set notation either by listing each of them using a comma or example $\left\{ 1,2,3,4,5 \right\}$ or using predicate. A predicate is a condition or rule that is either true or false. We use the predicate $P\left( x \right)$ to write that the set as $\left\{ x:P\left( x \right) \right\}$ or $\left\{ x|P\left( x \right) \right\}$ where the colon and the vertical bar repent such that.
We are asked to write $x> -17$ in set notation. So we have to write a set that represents all real numbers greater than $-17$. So we use the given condition $P(x):x >-17$ as predicate to write in set notation as
\[\left\{ x:P\left( x \right) \right\}=\left\{ x:x>-17,x\in \mathsf{\mathbb{R}} \right\}\]
Here $\mathsf{\mathbb{R}}$ is the set of real numbers. We know that we can write set of all numbers in between two numbers $a,b$ such that $a < b$ in interval form using square bracket $\left[ a,b \right]$ where both $a,b$ are included within the set and parenthesis $\left( a,b \right)$ where both $a,b$ excluded from the set. Here $a$ is called lower limit and $b$ is called upper limit. Mathematically we have
\[\begin{align}
  & \left[ a,b \right]=\left\{ x:a\le x\le b, x\in \mathsf{\mathbb{R}} \right\} \\
 & \left( a,b \right)=\left\{ x:a < x < b, x\in \mathsf{\mathbb{R}} \right\} \\
\end{align}\]
We are asked to write $x > -17$. So the lower limit of the interval is $a= -17$ and $a$ will not be included in the interval. The upper limit is infinity which means $b=\infty $ and we know that $\infty $ can never be included in the interval. So $x > -17$ in interval form is
\[x> -17=\left\{ x:x > -17,x\in R \right\}=\left( -17,\infty \right)\]

Note: We note that the real number is an ordered set which means if $x$ is a real number we can always find a real number greater or less than $x$. If the question would have asked for $\ge -17$ we would have included the lower limit $a=-17$ and would have written in interval from $\left[ -17,\infty \right)$. We should remember a square bracket is used for inclusion and a round bracket is used for exclusion of limits.