Question

The number of satisfying $| x - 2 |^{10x^2 - 1} = |x - 2|^{3x}$ is
A. 5
B. 6
C. 2
D. 4

Answer 1

Answer:

D. 4

Step-by-step explanation:

Without actually solving the equation, recall that for $a=|b|$, there are two cases:

$\begin{cases}a=b, \\a=-b\end{cases}$

In the given equation $|x-2|^{10x^2-1}=|x-2|^{3x}$, there are two pairs of absolute value symbols.

Since each has two cases, there must be a total of $2\cdot 2=\boxed{4}$ different equations created.

All four cases are:

$\begin{cases}(x-2)^{10x^2-1}=(x-2)^{3x},\\(-x+2)^{10x^2-1}=(x-2)^{3x},\\(x-2)^{10x^2-1}=(-x+2)^{3x},\\(-x+2)^{10x^2-1}=(-x+2)^{3x}\end{cases}$

Exponents differ, hence clearly there are four possible solutions to this equation.

You can solve for all four values of $x$ by taking the log of both sides and using a bit of algebra to verify you have four solutions.