Answer:
D. 4
Step-by-step explanation:
Without actually solving the equation, recall that for [tex]a=|b|[/tex], there are two cases:
[tex]\begin{cases}a=b, \\a=-b\end{cases}[/tex]
In the given equation [tex]|x-2|^{10x^2-1}=|x-2|^{3x}[/tex], there are two pairs of absolute value symbols.
Since each has two cases, there must be a total of [tex]2\cdot 2=\boxed{4}[/tex] different equations created.
All four cases are:
[tex]\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}[/tex]
Exponents differ, hence clearly there are four possible solutions to this equation.
You can solve for all four values of [tex]x[/tex] by taking the log of both sides and using a bit of algebra to verify you have four solutions.
