Question

Solutions for all numbers x ∈ R for the following equation?

The particular questions are:

`∣x+3∣+∣5+4x∣=16`

Answer 1

Answers are `x=8/5 and x= -24/5`

Explanation:

We have two modulus monomials added to be equal `16.`

It means that for every single monomial we will have two options :
when the expression inside is positive and when it is negative.

It means that overall we will have four different cases:

1. When `x+3>0 and 5+4x >0`
   so in this case, x has to be :`x> -3 and x> -5/4`

What this means is that x should be x > -5/4

when you solve the equation for this these conditions, you get
`x+3 + 5 + 4x = 16` where x=5/8 , which agrees with your condition that x has to be greater than `-5/4.`

You do in all cases the same process.

1. (The second case ) you have `x+3>0 and 5+4x <0`

`x> -3 and x <-5/4,` so x should be between -3 and -5/4

`-3 < x < -5/4`

when you solve `x+3 -(5+4x) = 16` you get that x = -6

`-6` is not between `-3 and -5/4` , so in the second case there is no solution

Two more cases you do on the same way.
You will get:
`3. x+3<0 and 5+4x <0`
`x=-24/5`

`4. x+3<0 and 5+4x >0`
no solution

So the possible solutions are only `x=5/8 and x = -24/5`

This can be done using graphical method also, but I prefer this one.