If x is a real number such that |x^2 - 9| = 5|x - 3|, which of the following could be the … — GRE General Test

If $x$ is a real number such that $|x^2 - 9| = 5|x - 3|$ , which of the following could be the value of $x$ ? Select all such values.

The values of

x

that satisfy the equation are

-8

2

, and

3

Factoring the left side of

|x^2 - 9| = 5|x - 3|

yields

|x - 3||x + 3| = 5|x - 3|

. Setting the common factor

|x - 3| = 0

gives

x = 3

. Dividing both sides by the non-zero quantity

|x - 3|

leaves

|x + 3| = 5

, which splits into

x + 3 = 5 \implies x = 2

and

x + 3 = -5 \implies x = -8

. Therefore,

-8

2

, and

3

are all valid solutions.

Apply the product rule for absolute values to factor the left-hand side.

|x^2 - 9| = |(x - 3)(x + 3)| = |x - 3| \cdot |x + 3|

, rewriting the equation as

|x - 3| \cdot |x + 3| = 5|x - 3|

The absolute value of a product is equal to the product of the individual absolute values.

Evaluate the case where the shared factor is zero:

|x - 3| = 0

x - 3 = 0 \implies x = 3

. Both sides equal

0

, making

x = 3

a valid solution.

Dividing by

|x - 3|

without checking if it can be zero would cause the loss of the root

x = 3

Evaluate the case where

|x - 3| \neq 0

by dividing both sides of the equation by

|x - 3|

|x + 3| = 5

Since

|x - 3| > 0

, we can safely divide both sides by this non-zero quantity.

Solve the remaining absolute value equation

|x + 3| = 5

x + 3 = 5 \implies x = 2

, and

x + 3 = -5 \implies x = -8

An expression inside an absolute value equal to a positive number

k

can equal either

k

-k

Factoring absolute value expressions using

|ab| = |a||b|

and systematically considering all cases to avoid dropping zero-roots or negative solutions.

Question