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Question

Part 1
Use algebraic​ and/or graphical methods to solve the inequality.
left parenthesis x minus 6 right parenthesis cubed less than 27
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Part 1
What is the solution of the​ inequality? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ solution(s) is/are xequals
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
B.
The solutions are the values of x in the interval
  
enter your response here.
​(Type your answer in interval​ notation.)
C.
There is no solution.

Answer 1

To solve the inequality \((x - 6)^3 < 27\), we can start by simplifying it.

1. Take the cube root of both sides: \[ x - 6 < 3 \] (since \(\sqrt[3]{27} = 3\))

2. Solve for \(x\): \[ x < 3 + 6 \] \[ x < 9 \]

Therefore, the solution to the inequality is that \(x\) can take any value less than 9.

Now, we can express the solution in the required format:

Choice B is appropriate since it asks for the values of \(x\) in interval notation.

The interval representing the solution is: \[ (-\infty, 9) \]

So the final answer is:

B. The solutions are the values of \(x\) in the interval \((- \infty, 9)\).