Solving an equation with absolute value and exponent

Published on November 12, 2024

Algebra

The equation (x-1)^(|x|-4) = 1 has solutions x = -4, 0, 2, and 4.

Question

\( \begin{array}{c}(x-1)^{|x|-4}=1 \\ A=-4 \\ B=0 \\ C=2 \\ D=4 \\ E=\text { all }\end{array} \)

Answer

We want to find the values of

that satisfy the equation

. There are three cases to consider:

Case 1: The base equals 1

, then

. Let’s check if this works in the original equation:

. So

is a solution.

Case 2: The exponent equals 0 and the base is not zero

, then

, which means

.
If

, then

. So

is a solution.
If

, then

. So

is a solution.

Case 3: The base equals -1 and the exponent is an even integer

, then

. Now we check the exponent:

, which is an even integer.
So,

. Thus,

is a solution.

Therefore, the solutions are

, and

. Comparing these to the given options, we have

, and

. All of these are solutions.

Final Answer: The final answer is

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