Difference between revisions of "2010 AMC 10B Problems/Problem 13"
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| Line 19: | Line 19: | ||
<math> | <math> | ||
2x-|60-2x|=x | 2x-|60-2x|=x | ||
| + | |||
x=|60-2x| | x=|60-2x| | ||
| + | |||
</math> | </math> | ||
| Line 26: | Line 28: | ||
<math> | <math> | ||
x=60-2x | x=60-2x | ||
| + | |||
3x=60 | 3x=60 | ||
| + | |||
x=20 | x=20 | ||
</math> | </math> | ||
| Line 41: | Line 45: | ||
<math> | <math> | ||
2x-|60-2x|=-x | 2x-|60-2x|=-x | ||
| + | |||
3x=|60-2x| | 3x=|60-2x| | ||
</math> | </math> | ||
| Line 48: | Line 53: | ||
<math> | <math> | ||
3x=60-2x | 3x=60-2x | ||
| + | |||
5x=60 | 5x=60 | ||
| + | |||
x=12 | x=12 | ||
</math> | </math> | ||
| Line 56: | Line 63: | ||
<math> | <math> | ||
-3x=60-2x | -3x=60-2x | ||
| + | |||
-x=60 | -x=60 | ||
| + | |||
x=-60 | x=-60 | ||
</math> | </math> | ||
Since an absolute value cannot be negative, we exclude <math>x=-60</math>. The answer is <math>20+60+12=92</math> | Since an absolute value cannot be negative, we exclude <math>x=-60</math>. The answer is <math>20+60+12=92</math> | ||
Revision as of 21:13, 24 January 2011
Problem
What is the sum of all the solutions of 
?
Solution
Case 1:
$2x-|60-2x|=x
x=|60-2x|$ (Error compiling LaTeX. Unknown error_msg)
Case 1a:
$x=60-2x
3x=60
x=20$ (Error compiling LaTeX. Unknown error_msg)
Case 1b:
Case 2:
$2x-|60-2x|=-x
3x=|60-2x|$ (Error compiling LaTeX. Unknown error_msg)
Case 2a:
$3x=60-2x
5x=60
x=12$ (Error compiling LaTeX. Unknown error_msg)
Case 2b:
$-3x=60-2x
-x=60
x=-60$ (Error compiling LaTeX. Unknown error_msg)
Since an absolute value cannot be negative, we exclude 
. The answer is 
