Question

How do you solve and write the following in interval notation: `(x + 6)(x + 8) ≥ 0`?

Answer 1

The solution is `x in (-oo,-8] uu [-6, +oo)`

Explanation:

Let `f(x)=(x+6)(x+8)`

Build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-8` `color(white)(aaaaaaa)` `-6` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+8` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x+6` `color(white)(aaaaa)` `-` `color(white)(aaaa)` ###color(white)(aaaaa)#`-` `color(white)(aa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `0` `color(white)(aaaa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+`

Therefore,

`f(x)>=0` when `x in (-oo,-8] uu [-6, +oo)`

graph{(x+6)(x+8) [-11.54, 0.947, -2.155, 4.09]}

Answer 2

`( -oo, -8 ] uu [ -6, oo )`

Explanation:

`( x + 6 )` is ZERO for `x = ( -6 )`

`( x + 6 )` is POSITIVE for `x > ( -6 )`

`( x + 8 )` is ZERO for `x = ( -8 )`

`( x + 8 )` is POSITIVE for `x > ( -8 )`

Refer to the SIGN Table

Our required Interval Notation can be written as follows:

`( -oo, -8 ] uu [ -6, oo )`