Question

How do solve `x^2>4` and write the answer as a inequality and interval notation?

Answer 1

`x<-2` or `x>2`
Interval notation:
`(-oo,-2)uu(2,oo)`

Explanation:

To solve, all we need to do is take the square root of both sides to get `x` by itself:
`sqrt(x^2)>sqrt(4)`

`|x|>2`

Which is true either when `x>2` or when `x<-2`

To express the answer in interval notation,we think about which interval satisfies this inequality. It's simply all numbers less than `-2`, and all numbers greater than `2`. This is the interval from `-oo` to `-2` and from `2` to `oo`. Note that the interval does not include `2` or `-2` themselves.

We can write the interval like so:
`(-oo,-2)uu(2,oo)`

The `uu` (`U` for union) symbol means to combine the two intervals.

Answer 2

`x < -2 vv x > 2`

`x in (-oo, -2) uu (2, oo)`

Explanation:

Given:

`x^2 > 4`

Subtract `4` from both sides to get:

`x^2-4 > 0`

Factor the left hand side to find:

`(x-2)(x+2) > 0`

The left hand side is positive if both of the factors are positive or both of the factors are negative.

Hence:

`x < -2" "` or `" "x > 2`

In interval notation:

`x in (-oo, -2) uu (2, oo)`