Respuesta :
Whenever you find an equation involving absolute values, you have to think that you're actually multiple different equations. The absolute values work as a switch, "activating" one of the various form of the equation.
In fact, the absolute value always returns the positive version of the quantity it holds, as in
[tex] |x| = \begin{cases} x &\text{if } x\geq 0\\ -x &\text{if } x < 0 \end{cases}[/tex]
In your case, this means that
[tex] |x-1| = \begin{cases} x-1 &\text{if } x-1\geq 0 \iff x\geq 1\\ -x+1 &\text{if } x < 1 \end{cases}[/tex]
and
[tex] |x-4| = \begin{cases} x-4 &\text{if } x-4 \geq 0 \iff x\geq 4\\ -x+4 &\text{if } x < 4 \end{cases}[/tex]
So, we can split the real numbers in three different zones: if [tex] x < 1 [/tex], both expressions are negative, so you have [tex] |x-1| = -x+1 [/tex] and [tex] |x-4| = -x+4 [/tex]. Otherwise, if [tex] 1\leq x<4 [/tex], [tex] x-1 [/tex] is positive, while [tex] x-4 [/tex] is still negative. This means that [tex] |x-1| = x-1 [/tex] and [tex] |x-4| = -x+4 [/tex]. Finally, if [tex] x\geq 4 [/tex], then both quantities are positive, and the absolute values won't change them: [tex] |x-1| = x-1 [/tex] and [tex] |x-4| = x-4 [/tex].
As you may imagine, we have three different equations, depending on which part of the real number line we're considering:
First zone: x < 1
Given everything we've said about how to solve absolute values, in this area the equation becomes
[tex] -x+1-x+4=3 \iff -2x+5=3 \iff -2x = -2 \iff x = 1 [/tex]
But we are assuming that [tex] x < 1 [/tex], so we can't accept this solution.
Second zone: [tex] 1 \leq x < 4 [/tex]
In this area the equation becomes
[tex] x-1-x+4=3 \iff 3 = 3 [/tex]
Which is always true. All numbers in [tex] [1,4) [/tex] are a solution of this equation.
Third zone: [tex] x \geq 4 [/tex]
In this area the equation becomes
[tex] x-1+x-4=3 \iff 2x - 5 = 3 \iff 2x = 8 \iff x = 4 [/tex]
So, 4 is also a solution. The complete set of solutions is given by [tex] [1,4) \cup \{4\} = [1,4] [/tex]
So, all numbers between 1 and 4, 1 and 4 included, are solutions of this equation.
