Question

How do you solve `(2x)/(x+5)<=0`?

Answer 1

`-5 < x <= 0`

Explanation:

`color(red)("There is a trap in this question in that there are excluded")` `color(red)("values for "x)`

`color(blue)("Step 1")`
Solve disregarding the excluded values.

Multiply both sides by `(x+5)`

`2x<=0xx(x+5)`

`2x<=0`

`color(blue)("Condition 1: "x<=0)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 2")`
An equation or expression becomes 'undefined' if you have division by 0.

`=>(x+5) != 0`

`color(blue)("Condition 2: "x!=-5)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 3")`

Can we have `x< -5color(white)(.)?`

Suppose `x=-6` then we have:

`(2(-6))/(-6+5) = (-12)/-1=+12`

This is not an excluded value but does not satisfy `(2x)/(x+5)<=0`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting it all together using proper notation")`

`-5 < x <= 0`