If a and b are real numbers such that |a - 3| = 5 and |2b + 1| = 9, what is the minimum po… — GRE General Test

If $a$ and $b$ are real numbers such that $|a - 3| = 5$ and $|2b + 1| = 9$ , what is the minimum possible value of $|a - b|$ ?

A
1
B
2
3Answer
D
4
E
6

Answer

The minimum possible value of

|a - b|

3

Solving

|a - 3| = 5

gives two possible values for

a

a = 8

and

a = -2

. Solving

|2b + 1| = 9

gives two possible values for

b

b = 4

and

b = -5

. Evaluating the distance

|a - b|

for all four pairs

(a,b)

gives

|8 - 4| = 4

|8 - (-5)| = 13

|-2 - 4| = 6

, and

|-2 - (-5)| = 3

. The minimum possible value is

3

Step-by-Step Solution

Solve the absolute value equation

|a - 3| = 5

for all possible values of

a

a - 3 = 5 \implies a = 8

a - 3 = -5 \implies a = -2

. Thus,

a \in \{-2, 8\}

An absolute value equation

|x| = k

splits into two linear equations:

x = k

and

x = -k

Solve the absolute value equation

|2b + 1| = 9

for all possible values of

b

2b + 1 = 9 \implies 2b = 8 \implies b = 4

2b + 1 = -9 \implies 2b = -10 \implies b = -5

. Thus,

b \in \{-5, 4\}

An absolute value equation

|2b + 1| = 9

has two cases:

2b + 1 = 9

and

2b + 1 = -9

Calculate

|a - b|

for all four possible pairs of

(a, b)

For

(8, 4): |8 - 4| = 4

.
For

(8, -5): |8 - (-5)| = 13

.
For

(-2, 4): |-2 - 4| = 6

.
For

(-2, -5): |-2 - (-5)| = 3

To find the minimum possible value of

|a - b|

, every valid combination of

a

and

b

must be tested.

Identify the minimum value among the calculated absolute differences.

The minimum calculated value is

3

Comparing

4, 13, 6,

and

3

yields

3

as the smallest value.

Key Concept

Solving absolute value equations and finding distances between points on the real number line

Estimated Time:1m 30s

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