|
|
| (6 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| − | == Problem ==
| + | #REDIRECT[[2003 AMC 12A Problems/Problem 6]] |
| − | Define <math>x \heartsuit y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y</math>. Which of the following statements is not true?
| |
| − | | |
| − | <math> \mathrm{(A) \ } x \heartsuit y = y \heartsuit x </math> for all <math>x</math> and <math>y</math>
| |
| − | | |
| − | <math>\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y) </math> for all <math>x</math> and <math>y</math>
| |
| − | | |
| − | <math>\mathrm{(C) \ } x \heartsuit 0 = x </math> for all <math>x</math>
| |
| − | | |
| − | <math>\mathrm{(D) \ } x \heartsuit x = 0 </math> for all <math>x</math>
| |
| − | | |
| − | <math> \mathrm{(E) \ } x \heartsuit y > 0 </math> if <math>x \neq y</math>
| |
| − | | |
| − | == Solution ==
| |
| − | Examining statement C:
| |
| − | | |
| − | <math> x \heartsuit 0 = |x-0| = |x| </math>
| |
| − | | |
| − | <math>|x| \neq x</math> when <math>x<0</math>, but statement D 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>
| |
| − | | |
| − | == See Also ==
| |
| − | *[[2003 AMC 10A Problems]]
| |
| − | *[[2003 AMC 10A Problems/Problem 5|Previous Problem]]
| |
| − | *[[2003 AMC 10A Problems/Problem 7|Next Problem]]
| |
| − | | |
| − | [[Category:Introductory Algebra Problems]]
| |
