2022 AMC 10A Problem 6

Question

Which expression is equal to

\left|a-2-\sqrt{(a-1)^2}\right|

for

a < 0?

3 - 2a

1 - a

1

a + 1

3

Replace the square root of a square by an absolute value

Use the sign of each expression when a is negative

By the definition of square root, we get that

\sqrt{(a - 1)^2} = |a - 1|.

Since

a < 0,

we get that

a - 1 < 0,

which means that

|a - 1| = 1 - a.

The whole expression therefore simplifies to

|a - 2 - (1 - a)| = |2a - 3|.

Since

a < 0,

we know that

2a - 3 < 0.

This means that

|2a - 3| = 3 - 2a.

Thus, A is the correct answer.

Problem 6 in other years