1973 AMC 12 Problem 22

Attempt Problem 22 of the 1973 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 1973 AMC 12 solutions, or check the answer key.

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22.

The set of all real solutions of the inequality

|x-1|+|x+2|\lt5

\{x:-3\lt x\lt2\}

\{x:-1\lt x\lt2\}

\{x:-2\lt x\lt1\}

\left\{x:-\dfrac32\lt x\lt\dfrac72\right\}

\varnothing

(empty)

Interpret the two absolute values as the distances from

x

1

and to

-2

Between

-2

and

1

the sum is constant; outside that interval it increases by twice the distance from the nearer endpoint

For

-2\le x\le1,

the sum of the distances from

x

-2

and

1

3.

x

lies a distance

u

outside this interval, the sum is

3+2u.

Thus

3+2u\lt5 \quad\Longleftrightarrow\quad u\lt1.

Extending the interval

[-2,1]

1

at each end gives

-3\lt x\lt2.

Therefore, the correct answer is A.

Problem 22 in other years