1985 AMC 12 Problem 8

Attempt Problem 8 of the 1985 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 1985 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Let

a,

a',

b,

b'

be real numbers with

a

and

a'

nonzero. The solution to

ax+b=0

is less than the solution to

a'x+b'=0

if and only if

a'b\lt ab'

ab'\lt a'b

ab\lt a'b'

\frac ba\lt\frac{b'}{a'}

\frac{b'}{a'}\lt\frac ba

Write each equation’s solution without cross-multiplying

Multiplying an inequality by

-1

reverses its direction

The two solutions are

-\frac{b}{a}

and

-\frac{b'}{a'}.

Thus the given comparison is

-\frac ba\lt-\frac{b'}{a'}.

Multiplying both sides by

-1

reverses the inequality and gives

\frac{b}{a}\gt \frac{b'}{a'},

equivalently

\frac{b'}{a'}\lt \frac{b}{a}.

No multiplication by the unknown-sign quantity

aa'

is valid.

Thus the correct answer is E.

Problem 8 in other years