The symbol |a| means +a if a is greater than or equal to zero, and -a if a is less than or equal to zero; the symbol means “greater than.” The set of values x satisfying the inequality |3-x|<4 consists of all x such that:

1959 AMC 12 Problem 25

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25.

The symbol

|a|

means

+a

a

is greater than or equal to zero, and

-a

a

is less than or equal to zero; the symbol

\lt

means “less than”; the symbol

\gt

means “greater than.” The set of values

x

satisfying the inequality

|3-x|\lt4

consists of all

x

such that:

x^2\lt49

x^2\gt1

1\lt x^2\lt49

-1\lt x\lt7

-7\lt x\lt1

Rewrite

|3-x|\lt4

as a compound inequality

Solve

-4\lt3-x\lt4

for

x

The absolute-value inequality is equivalent to

-4\lt3-x\lt4.

Subtracting

3

and multiplying by

-1

reverses the inequalities, giving

-1\lt x\lt7.

Thus, the correct answer is D.