Difference between revisions of "2003 AMC 10A Problems/Problem 6"
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<math> x \heartsuit 0 = |x-0| = |x| </math> | <math> x \heartsuit 0 = |x-0| = |x| </math> | ||
| − | <math>|x| \neq x</math> when <math>x<0</math>, but statement | + | <math>|x| \neq x</math> when <math>x<0</math>, but statement C says that it does for all <math>x</math>. |
Therefore the statement that is not true is "<math>x \heartsuit 0 = x</math> for all <math>x</math>" <math>\Rightarrow C</math> | Therefore the statement that is not true is "<math>x \heartsuit 0 = x</math> for all <math>x</math>" <math>\Rightarrow C</math> | ||
Revision as of 21:08, 24 March 2009
Problem
Define 
to be 
for all real numbers 
and 
. Which of the following statements is not true?
for all and
for all and
for all
for all
if 
Solution
Examining statement C:
when 
, but statement C says that it does for all .
Therefore the statement that is not true is "
for all " 
Alternatively, consider that the given "heart function" is actually the definition of the distance between two points. Examining all of the statements, only C is not necessarily true; if c is negative, the distance between 
and 
is the absolute value of , not itself, because distance is always nonnegative.
See Also
| 2003 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
