Question 230700: find all x in R for which :
1) absolute value of (x-1)<1/2
2) absolute value of (x^2-1)<1/2 Found 2 solutions by stanbon, jsmallt9:Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! find all x in R for which :
1) absolute value of (x-1)<1/2
-1/2 < x-1 < 1/2
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Add 1 along the line to get:
1/2 < x < 3/2
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2) absolute value of (x^2-1)<1/2
-1/2 < x^2-1 < 1/2
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Add 1 along the line to get:
1/2 < x^2 < 3/2
Take the square root along the line:
[sqrt(2)/2 < x < sqrt(6)/2
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Cheers,
Stan H.
You can put this solution on YOUR website! First let's review the idea of absolute value. The absolute value of a number is its distance from zero on the number line, without regard to direction.
Your first inequality says that
This says "the distance of x-1 from zero is less than 1/2". Now picture a number line and picture the numbers that would be within 1/2 of 0. Now, how do we describe these numbers in the form of inequalities. I hope the following makes sense. It is saying "x-1" is between -1/2 and 1/2": and
Notice that the absolute values are gone. This is the key step in solving absolute value problems: learning how to remove the absolute values by writing equivalent inequalities (or equations). Now we just solve these inequalities by adding 1 to each side of each inequality: and
This describes the solution set: "All numbers between 1/2 and 3/2 (not including 1/2 and 3/2)"
This one says that the distance of is less than 1/2. Using the same logic as above to rewrite it without absolute values we get: and
Since there is no x term (just terms, we'll isolate the squared terms" and
Now we'll find the square root of each side: and
Rationalizing the denominators we get: and
So our solution to this problem is "all numbers between and (exclusive of and ).
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