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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: $$(-\mathrm{\infty },9)$$

So the final answer is:

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