Question

How do you solve `|t + 4| > 10`?

Answer 1

`tin(-∞,-14)`U`(6,∞)`

Explanation:

Using the definition of the absolute value, we are able to split this inequality with absolute values into two without absolute values:

`t+4>10`, when `t+4>=0`
`-(t+4)>10`, when `t+4<0`

When solving each inequality separately, we get two conditions for t:

`t+4>10`
`(t+4)-4>10-4`
`t>6`

`-(t+4)>10`
`-t-4>10`
`(-t-4)+4>10+4`
`-t>14`
`-(-t)> -(14)`
`t<-14`

We must then intersect each condition with its corresponding restriction:

`(t>6)` ∩ `(t+4>=0)`
`(t>6)` ∩ `(t>=-4)`
`t>6`

`(t<-14)` ∩ `(t+4<10)`
`(t<-14)` ∩ `(t<-4)`
`t<-14`

Lastly, we must union these two new conditions:

`(t>6)` U `(t<-14)`
`tin(-∞,-14)`U`(6,∞)`

Answer 2

Read below.

Explanation:

If you have `absx>b`, then you can set it up as:

`x>b` or `x<-b`

Therefore,

`abs(t+4)>10` becomes `t+4>10` or `t+4<-10`

We solve each inequality.

`t+4>10`

`t>6`

`t+4<-10`

`t<-14`

`t` either has to be greater than `6` or less than `-14`.