Question

How do you solve and write the following in interval notation: `1/|x+5| >2`?

Answer 1

`-11/2 < x < -5` and `-5 < x < =9/2`

Explanation:

First of all, we have to exclude the value when the inequality is undefined because the denominator is zero:
`x != -5`

In all cases, except this, `|x+5| > 0`.

Both sides of an inequality can be multiplied by a POSITIVE number, leaving the sign of inequality as is.

Since `|x+5| > 0` (after we have excluded value `x=-5`), let's multiply left and right sides of our inequality by `|x+5|` getting
`(1 * |x+5|) / (|x+5|) > 2 * |x+5|`
or
`1 >2*|x+5|`

Next transformation is division of both sides of inequality by POSITIVE number `2` preserving the sign of inequality:
`1/2 > |x+5|`

Recall the definition of `|a|`:
`|a| = a` for `a >= 0` and
`|a| = -a` for `a < 0`

Consider now two cases (we excluded `x=-5`):

Case 1. `x+5 > 0` or, equivalently, adding `-5` to both parts of inequality preserving the sign of inequality, `x > -5`
Then `|x+5| = x+5` and our inequality looks like
`1/2 > x+5`
or, subtracting `5` from both sides and retaining the sign of inequality, `x < -9/2`
We have to combine this with an inequality that defines our case,
`x > -5`.
Both inequalities result in'
`-5 < x < -9/2`

Case 2. `x+5 < 0` or, equivalently, adding `-5` to both parts of inequality preserving the sign of inequality, `x < -5`
Then `|x+5| = -x-5` and our inequality looks like
`1/2 >- x-5`
or, add `x` to both sides and retaining the sign of inequality, `x +1/2 > -5`
or, subtracting `1/2` from both sides,
`x > -11/2`
We have to combine this with an inequality that defines our case,
`x < -5`.
Both inequalities result in'
`-11/2 < x < -5`

Final solution is
`-11/2 < x < -5` and `-5 < x < =9/2`

Here is an illustrative graph of a function `y= 1/|x+5|-2`. Notice, where it is positive.
graph{1/|x+5|-2 [-10, 2, -5, 5]}