I need to review modulus function

[sania123]

Can anyone help me understand this questions from 800score.com ? I stumbled upon this questions when i was reviewing the concepts for modulus function.

Solve
4 - 8|x - 1| = 5

http://800score.com/guidec6view1b.html

Kindly help me out on this

Thank You.

[Guest]

My review for the question from 800score.com is as follows:
For understanding the above problem the modulus function should be clear. You can refer in detail regarding the modulus here http://800score.com/guidec6view1b.htmlfunction

4-8|x-1| = 5

The modulus function can take both positive and negative values.
When the modulus function is positive

4-8(x-1) = 5
-8(x-1) = 1
x-1 = -1/8
x = 7/8

When the modulus function is negative

4+8(x-1) = 5
8(x-1) = 1
x-1 = 1/8
x = 9/8

So, the values of x are 7/8 and 9/8

Hope it helps.

Best of luck!!

[Guest]

The above reply is almost right but actually wrong (there are in fact no solutions).

Split the problem into two cases, according to the inside of the modulus function.

Case 1: [tex]x-1\geq 0[/tex]
Because [tex]x-1\geq 0[/tex] we know [tex]|x-1| = x-1[/tex], so we can use this to get rid of the modulus signs and solve for [tex]x[/tex]. So
[tex]4 - 8|x-1| = 5[/tex]
becomes
[tex]4 - 8(x-1) = 5[/tex]
Solving gives [tex]x=7/8[/tex] as in the previous post. But now you have to check if the initial assumption is satisfied, i.e. is [tex]x-1\geq 0[/tex], and unfortunately in this case it is not. So [tex]x=7/8[/tex] is not a solution, and there are no solutions when [tex]x-1\geq 0[/tex].

Case 2: [tex]x-1\leq 0[/tex]
Just as before we can use the assumption to get rid of the modulus signs as we know [tex]|x-1| = -(x-1)[/tex]. Solving for [tex]x[/tex] gives [tex]x=9/8[/tex], but this does not satisfy our initial assumption that [tex]x-1\leq 0[/tex], so again there are no solutions.

Consequently there are no solutions to the equations as either case 1 or 2 must hold and in either case there is no [tex]x[/tex] which satisfies the equation.

As a side note we could have seen quite easily that there are no solutions by observing that [tex]4-8|x-1|=5[/tex] rearranges to [tex]|x-1|=-1/8[/tex] and since [tex]|x-1|\geq 0[/tex] there is no way the equation could hold.

Hope this helped,

R. Baber.

[sania123]

Thank you for the answer and clearing my doubts.