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22 votes
22 votes
The number of
x satisfying
| x - 2 |^(10x^2 - 1) = |x - 2|^(3x) is

A. 5
B. 6
C. 2
D. 4

User Busta
by
2.6k points

1 Answer

17 votes
17 votes

Answer:

D. 4

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.

User Jakub Judas
by
3.3k points