Question

How do you solve and write the following in interval notation: `2x - 4< 4` or`x + 5 ≤ 2 - 2x`?

Answer 1

1. `x<4` and in interval notation `x in(-oo,4)`
2. `x<=-1` and in interval notation `x in(-oo,-1]`

Explanation:

Lets take the first example.

`2x-4<4`

This is an inequality. Inequalities are usually solved like regular equations.

Lets add `4` to both sides of the equation.

`2x-4+color(red)4<4+color(red)4`

`2x<color(red)8`

Divide both sides by `2`

`(2x)/color(red)2<8/color(red)2`

`x<4`

This inequality means that the value of `x` has to be less than `4` to satisfy the inequality.

In interval notation it will be written as `x in(-oo,4)` because the value of `x` can be any number between `-oo` and `4` but cannot be `-oo` or `4` hence we use this ( ) bracket.

Similarly when we simplify `x+5<=2-2x` we get `x<=-1`

So here the interval notation is `x in(-oo,-1]` because `x` can be any number between `-oo` and `-1` and can be `-1` but cannot be `-oo` hence this ( ] bracket.