# Why do absolute value equations have to have 2 answers?

• Sep 14th 2011, 03:05 PM
cytotoxictcell
Why do absolute value equations have to have 2 answers?
like this

[x-5] = 10

x-5=10 OR x-5=-10

x=15 x=-5

I am taking a web course so i dont have the benefit of the lecture =(,
• Sep 14th 2011, 03:23 PM
pickslides
Re: Why do absolute value equations have to have 2 answers?
Look at the functions y = |x-5| and y= 10 how many times do they intersect?
• Sep 14th 2011, 03:27 PM
Re: Why do absolute value equations have to have 2 answers?
Remember that an absolute value leaves positive numbers unchanged and multiplies negative numbers by (-1). Examples,
|5|=5, but |-5|=(-1)*-5=5.

In your question, you have an unknown expression in the absolute value; |x-5|=10, where x is a variable. Since the absolute value is equal to 10, knowing that it leaves positive numbers unchanged and makes negative numbers positive, you know that the inside of the absolute value, namely (x-5) is either 10 or -10, because |10|=10=|-10|.

You therefore have two different solutions for x, x=15 and x=-5, because the inside of the absolute value with these values will give you respectively 10 and -10.
• Sep 14th 2011, 03:37 PM
Plato
Re: Why do absolute value equations have to have 2 answers?
Quote:

Originally Posted by cytotoxictcell
like this

[x-5] = 10

x-5=10 OR x-5=-10

x=15 x=-5

I am taking a web course so i dont have the benefit of the lecture =(,

Please learn to use LaTeX tags.
[TEX]|x-5|=10[/TEX] gives $\displaystyle |x-5|=10$

To understand the question you asked one must understand what $\displaystyle |a|$ really means.
If stands for the distance that the number $\displaystyle a$ is from $\displaystyle 0$.

Thus $\displaystyle |6|=6$ because $\displaystyle 6$ is six units from zero.

$\displaystyle |-6|=6$ because $\displaystyle -6$ is also six units from zero.

So $\displaystyle |x|=10$ means that $\displaystyle x=10\text{ or }x=-10.$

If you have $\displaystyle |x-5|=10$ then you must realize that $\displaystyle x-5$ is just some number.

So $\displaystyle |x-5|=10$ means that $\displaystyle x-5=10\text{ or }x-5=-10.$
• Sep 15th 2011, 03:34 AM
HallsofIvy
Re: Why do absolute value equations have to have 2 answers?
By the way, although absolute value equations may have two solutions, they don't have to have have two solutions. The absolute value equation |x|= 0 has the single solutions x= 0.