# How do you solve and write the following in interval notation: y <3 or 3y + 4 > -5?

Jan 18, 2018

$\left(- \infty , 3\right) \cup \left(- 3 , \infty\right)$

#### Explanation:

Let's solve this one component at a time

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

$y < 3$ is saying that $y$ is less than -- but not including -- $3$. So we just need to convey that same information in interval notation.

First, do we use $\left(\textcolor{w h i t e}{.}\right)$ or $\left[\textcolor{w h i t e}{.}\right]$? Well, we aren't including $3$, so we use $\left(\textcolor{w h i t e}{.}\right)$.

Now, what do we do with the $3$? Well, are we starting at $3$ or is it our end point? Are we saying that we go from $3$ to infinity or from (negative) infinity to $3$? In our case, we're going from infinity to $3$ and $3$ is our greatest value. So since we stop at $3$, we put $3$ last in the parentheses.

$\left(- \infty , 3\right)$

color(white)

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

Now let's do the other one

$3 y + 4 > - 5$

first, we ought to get $y$ on the right side of the sign:

$3 y > - 9$

$y > - 3$

Now we go through the same process as before:

$\left(\textcolor{w h i t e}{.}\right)$ or $\left[\textcolor{w h i t e}{.}\right]$?

Not including $- 3$ so we'll use $\left(\textcolor{w h i t e}{.}\right)$

This says that $y$ is larger than $- 3$, or that $y$ goes from $- 3$ to infinity

$\left(- 3 , \infty\right)$

$\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

Now we put these together. it's pretty simple. We just need to figure out if we use $\cup$ or $\cap$?

I like to remember $a \cap d$ ( and ). Well in our case, we were told "or", so we should use $\cup$

Our final answer is $\left(- \infty , 3\right) \cup \left(- 3 , \infty\right)$