1990 AMC 12 Problem 8

Attempt Problem 8 of the 1990 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1990 AMC 12 solutions, or check the answer key.

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The number of real solutions of the equation

|x-2|+|x-3|=1

0

1

2

3

more than

3

Interpret the two absolute values as distances on a number line

Consider every

x

between

2

and

3

For every

x\in[2,3],

\begin{aligned} &|x-2|+|x-3|\\ &\quad=(x-2)+(3-x)\\ &=1. \end{aligned}

Thus the entire interval

[2,3]

consists of solutions, so there are more than

3.

Thus the correct answer is E.

Problem 8 in other years