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2014 AMC 10A Problem 7

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2014 AMC 10A Problem 7AMC 10 Step by Step
Name:                             Date: September 13, 2026
Absolute Value & Inequalities · difficulty 2 of 5 · about 3 min · position P1-10

Nonzero real numbers xx , yy , aa , and bb satisfy x<ax < a and y<by < b . How many of the following inequalities must be true?

(I) x+y<a+b\textbf{(I)}\ x+y < a+b\qquad

(II) x−y<a−b\textbf{(II)}\ x-y < a-b\qquad

(III) xy<ab\textbf{(III)}\ xy < ab\qquad

(IV) xy<ab\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}

  1. A)

    00

  2. B)

    11

  3. C)

    22

  4. D)

    33

  5. E)

    44

Answer

B

Key insight

Only addition preserves both inequalities; a single negative example such as x = y = -2, a = b = -1 breaks II, III and IV at once.

Solution

(I) Inequalities in the same direction can be added: x+y<a+bx + y < a + b. This one is always true.

(II) Subtracting reverses the effect of the second inequality. Take x=1,a=2x = 1, a = 2 and y=1,b=10y = 1, b = 10: then x−y=0x - y = 0 but a−b=−8a - b = -8, so II fails.

(III) Products misbehave with negatives. Take x=y=−2x = y = -2 and a=b=−1a = b = -1: both hypotheses hold, but xy=4xy = 4 and ab=1ab = 1, so III fails.

(IV) The same example gives xy=1\frac{x}{y} = 1 and ab=1\frac{a}{b} = 1, so the strict inequality fails.

Exactly one statement must be true.

The answer is (B) 1\boxed{\textbf{(B)}\ 1}.

Why this works

"Must be true" means true for all allowed values, so one counterexample is enough to reject a statement. The only operation that always respects two inequalities in the same direction is addition. Subtraction, multiplication and division all depend on signs and magnitudes, and negative test values expose that quickly. Keep a favorite negative example (like −2<−1-2 < -1) ready for inequality problems.

The trap

Assuming the variables are positive, which makes III look true and leads to answering 2 or 3.

Common mistakes

  • Assuming the variables are positive, which makes III look true and leads to answering 2 or 3.
  • Believing II must be true because "subtracting a smaller from a smaller"; the sign of the yy terms flips.

Techniques

Test small/specific values or special cases to find or verify the answer