Express w/o absolute values: (a) | 2(-5/2) |, (b) | x | < 10, (c) | 2 - sqrt{5} |

hollym123

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hiya i have this homework question and was wondering if I am doing it right.

question a. I just simplified the equation so it turned into -10/2 then -5. is that the answer? so then for b) what would I do for b? and c? a bit confused thanks 11161
 

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Let's begin with the definition:

[MATH]|x|=\begin{cases}-x, & x<0 \\[3pt] x, & 0\le x \\ \end{cases}[/MATH]
You have correctly determined the value of the expression within the absolute value brackets in part a). Using the definition above, can you now state the value of the complete expression (including the absolute value)?
 
Let's begin with the definition:

[MATH]|x|=\begin{cases}-x, & x<0 \\[3pt] x, & 0\le x \\ \end{cases}[/MATH]
You have correctly determined the value of the expression within the absolute value brackets in part a). Using the definition above, can you now state the value of the complete expression (including the absolute value)?
would it be 5 < 0 ?
 
No, since -5 < 0 then by the definition I gave:

[MATH]|-5|=-(-5)=5[/MATH]
Before we move on, does that make sense to you?
 
Okay, let's look at b):

[MATH]|x|<10[/MATH]
This tells us we want the set of all real numbers whose magnitude, or equivalently, whose distance from zero on the number line is less than 10. Can you rewrite this inequality as a compound inequality without an absolute value?
 
There is no need to multiply and divide by 2. They cancel out! Did you notice that when you multiplied -5 by 2 and then divide by 2 you got back -5? One way of computing the absolute value of a single number is to write down the number within the bars the bars without a sign. S0 | -5 | = 5.

For b think of which numbers when you ignore the sign will be less than 10. Remember that there are many many numbers between 9 and 10

Note 2 = sqrt(4) < sqrt(5) < sqrt(9) = 3, ie sqrt(5) is between 2 and 3. So 2-sqrt(5) <0. Hence sqrt(5) - 2 >0. So |2-sqrt(5)| = 2-sqrt(5)
 
To follow up...part b)

[MATH]-10<x<10[/MATH]
c) [MATH]|2-\sqrt{5}|=-(2-\sqrt{5})=\sqrt{5}-2[/MATH]
 
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