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

The particular questions are:

∣x+3∣+∣5+4x∣=16

1 Answer
Mar 9, 2018

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.