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Difference between revisions of "2004 AMC 10A Problems/Problem 4"

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==Solution==
 
==Solution==
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<math>|x-1|</math> is equivalent to the distance between <math>x</math> and <math>1</math>; <math>|x-2|</math> is equivalent to the distance between <math>x</math> and <math>2</math>.
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<math>|x-1|</math> is the distance between <math>x</math> and <math>1</math>; <math>|x-2|</math> is the distance between <math>x</math> and <math>2</math>.
  
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Therefore, <math>x</math> is equidistant from <math>1</math> and <math>2</math>, so <math>x=\frac{1+2}2=\frac32\Rightarrow\mathrm{(D)}</math>.
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Therefore, the given equation says <math>x</math> is [[equidistant]] from <math>1</math> and <math>2</math>, so <math>x=\frac{1+2}2=\frac32\Rightarrow\mathrm{(D)}</math>.
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Alternatively, we can solve by casework (a method which should work for any similar problem involving [[absolute value]]s of [[real number]]s).  If <math>x \leq 1</math> then <math>|x - 1| = 1-x</math> and <math>|x - 2| = 2 - x</math> so we must solve <math>1 - x = 2 - x</math> which has no solutions.  Similarly, if <math>x \geq 2</math> then <math>|x - 1| = x - 1</math> and <math>|x - 2| = x - 2</math> so we must solve <math>x - 1 = x- 2</math>, which also has no solutions.  Finally, if <math>1 \leq x \leq 2</math> then <math>|x - 1| = x - 1</math> and <math>|x - 2| = 2-x</math> so we must solve <math>x - 1 = 2 - x</math>, which has the unique solution <math>x = \frac32</math>.
  
 
==See Also==
 
==See Also==

Revision as of 14:06, 12 November 2006

Problem

What is the value of $x$ if $|x-1|=|x-2|$?

$\mathrm{(A) \ } -\frac12 \qquad \mathrm{(B) \ } \frac12 \qquad \mathrm{(C) \ } 1 \qquad \mathrm{(D) \ } \frac32 \qquad \mathrm{(E) \ } 2$

Solution

$|x-1|$ is the distance between $x$ and $1$; $|x-2|$ is the distance between $x$ and $2$.

Therefore, the given equation says $x$ is equidistant from $1$ and $2$, so $x=\frac{1+2}2=\frac32\Rightarrow\mathrm{(D)}$.

Alternatively, we can solve by casework (a method which should work for any similar problem involving absolute values of real numbers). If $x \leq 1$ then $|x - 1| = 1-x$ and $|x - 2| = 2 - x$ so we must solve $1 - x = 2 - x$ which has no solutions. Similarly, if $x \geq 2$ then $|x - 1| = x - 1$ and $|x - 2| = x - 2$ so we must solve $x - 1 = x- 2$, which also has no solutions. Finally, if $1 \leq x \leq 2$ then $|x - 1| = x - 1$ and $|x - 2| = 2-x$ so we must solve $x - 1 = 2 - x$, which has the unique solution $x = \frac32$.

See Also

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