Let a and b be non-zero real numbers such that |a + b| = 3|a - b|. What is the value of \l… — GRE General Test

Let $a$ and $b$ be non-zero real numbers such that $|a + b| = 3|a - b|$ . What is the value of $\left|\frac{a^2 + b^2}{ab}\right|$ ?

The value of

\left|\frac{a^2 + b^2}{ab}\right|

\frac{5}{2}

Squaring both sides of the given equation

|a + b| = 3|a - b|

eliminates the absolute values to give

(a + b)^2 = 9(a - b)^2

. Expanding and simplifying yields

2a^2 - 5ab + 2b^2 = 0

, which factors as

(2a - b)(a - 2b) = 0

. Thus,

b = 2a

a = 2b

. In either case, substituting into

\left|\frac{a^2 + b^2}{ab}\right|

reduces the expression to

\frac{5}{2}

Square both sides of the absolute value equality

(a + b)^2 = 9(a - b)^2

Since both sides of

|a + b| = 3|a - b|

are non-negative real numbers, squaring preserves equality and eliminates the absolute value signs.

Expand both algebraic expressions and collect like terms

a^2 + 2ab + b^2 = 9a^2 - 18ab + 9b^2 \implies 8a^2 - 20ab + 8b^2 = 0 \implies 2a^2 - 5ab + 2b^2 = 0

Expanding the binomial squares allows combining like terms to solve for the relationship between

a

and

b

Factor the quadratic expression in terms of

a

and

b

(2a - b)(a - 2b) = 0 \implies b = 2a \text{ or } a = 2b

Factoring determines the exact proportional relationship between

a

and

b

Substitute the relation into the target expression

\left|\frac{a^2 + (2a)^2}{a(2a)}\right| = \left|\frac{5a^2}{2a^2}\right| = \frac{5}{2}

Substituting

b = 2a

allows

a^2

to cancel completely, leaving a constant numerical value.

Properties of Real Numbers and Absolute Value Equations

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