AMC 10 · 2004 · #4

Grade 7 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

absolute-valuelinear-equations-one-var physical-representationconvert-to-algebra ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 1 insight

Problem

x

is a real number that satisfies the equation

|x-1|=|x-2|

. What is the value of

x

Pick an answer.

(A)

$-\frac12$

(B)

$\frac12$

(C)

(D)

$\frac32$

(E)

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The two absolute values are begging to be read as distances on a number line (Tool #1): |x-1| is how far x sits from 1, and |x-2| is how far it sits from 2. Wanting those distances equal turns the algebra into a one-line geometry fact — the point equidistant from 1 and 2 is the point halfway between them. Tool #6 (Guess and Check) then confirms the candidate by plugging it back in, and Tool #4 (Introduce a Variable) offers a purely algebraic cross-check by squaring both sides.

1STEP 1

Read each side as a distance

On a number line $|x-1|$ is the distance from $x$ to 1 and $|x-2|$ the distance to 2, so $x$ sits equally far from both.

|x-1| = dist(x,1), |x-2| = dist(x,2)

2STEP 2

The equidistant point is the midpoint

A point equally far from 1 and 2 must be their midpoint, the average: $x=\frac{1+2}{2}=$ $\frac{3}{2}$ .

x = (1+2)/2 = 3/2

3STEP 3

Check it and name the choice

Check both sides: $\left|\frac{3}{2}-1\right|=\frac{1}{2}=\left|\frac{3}{2}-2\right|$ , so the choice is (D).

|3/2-1| = 1/2 = |3/2-2| → (D)

Answer

3/2

The solution must lie strictly between 1 and 2, since only a point inside that gap can be equally far from both endpoints. That instantly rules out (A) -1/2, (B) 1/2, (C) 1, and (E) 2, all of which sit on or outside the ends. Only (D) 3/2 lands in the middle, and 3/2=1.5 is exactly halfway, so the answer is reasonable.

💡Key takeaway

When a point is the same distance from two numbers on the line, it sits exactly halfway between them — the average of the two.

Read each side as a distance
The equidistant point is the midpoint
Check it and name the choice

Read each side as a distance

The equidistant point is the midpoint

Check it and name the choice

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