Does anyone can solve the following equation?
|x − 1| = 1 − x
Yes I can. Thanks for inquiring!
Sorry, that's unfair - it's your first post. Can you show us the work you have so far? Do you understand what absolute value means?
Square both sides. And see what you get.
A more introductory way is to consider (x -1) as separately positive, and negative.
If x - 1 is positive, then x - 1 = 1 - x;
If x - 1 is negative, then x - 1 = -(1 - x )
Actually, that is not an inequality question; but an absolute value equation.
If you have |a| = -a, what is the only way this can be true?
This is the solutions which is done by myself:
|x − 1| = 1 − x
x − 1 = ±(1 − x)
x − 1 = +(1 − x)
= 1 − x
x − 1 = −(1 − x)
= −1 + x
= x − 1
By |x| = ±x properties.
That's absolutely correct. I mean ABSOLUTE VALUE QUESTION. Thanks for remind me about that.
Why should I square-ing both sides of them? This is the most question I want to know in the ABSOLUTE VALUE. As I can read on Calculus book, there is no rules about SQUARE on absolute value except the |x| = √x^2 one. Can you explain me further about this?
tomcenjerrym - Your first condition yields x=0 as a solution; and your second condition allows ALL real numbers as solutions. All real numbers will satisfy the equation.
No they won't. Take x=5 as an example |5-1|=|4|=4, but 1-5=-4, so in this case |x-1| does not equal 1-x, hence it is obviously not true for all real numbers. There is, however, a subset of the real numbers (with more than a single element) that satisfies the above equation.
You're correct. I was not careful enough when I solved the problem. We must watch around the critical point. The first part indicates x=1. When we check a value less than 1, we find equality; when we check a point greater than 1, we do not find equality.
The solution seems to be x<=1
I like nicktacik way to solve the problem. It’s simple and easy to understand. Thank you very much nicktacik and thank you to everyone too.
The title "Inequality Question" is thus not entirely wrong, since the solution is an inequality.
ya, his answer was kind of making me suddenly awake!
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