If a and b are nonzero integers such that |a| < |b| and a \cdot b < 0, which of the follow… — ACT

If $a$ and $b$ are nonzero integers such that $|a| < |b|$ and $a \cdot b < 0$ , which of the following expressions must have the greatest value?

The expression

|a - b|

must have the greatest value because

a

and

b

have opposite signs, making

|a - b| = |a| + |b|

Because

a

and

b

have opposite signs (

a \cdot b < 0

), the distance between them on the number line is the sum of their individual distances from zero,

|a| + |b|

. The expression

|a - b|

measures this exact distance, which is guaranteed to be greater than any net difference or signed value.

Analyze the given conditions

Since

a \cdot b < 0

a

and

b

have opposite signs (one positive, one negative). Since

|a| < |b|

, the absolute value of

b

is strictly greater than the absolute value of

a

Determining the signs and relative magnitudes of

a

and

b

is necessary to evaluate absolute value expressions.

Evaluate

|a - b|

in terms of magnitudes

When two numbers have opposite signs, subtracting them adds their magnitudes:

|a - b| = |a| + |b|

a > 0

and

b < 0

|a - b| = a - (b) = |a| + |b|

. If

a < 0

and

b > 0

|a - b| = |- |a| - |b|| = |a| + |b|

Compare

|a - b|

to the other expressions

|a - b| = |a| + |b|

is strictly greater than

|a + b| = |b| - |a|

|b| - |a|

|a| - |b|

, and

a - b

Since both

a

and

b

are non-zero,

|a| + |b| > |b| - |a| > 0 > |a| - |b|

Absolute value distance and sign properties on a number line

Question