Question

Use algebraic​ and/or graphical methods to solve the inequality.
left parenthesis x minus 9 right parenthesis cubed less than 64
Question content area bottom
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 \((x - 9)^3 < 64\), we first rewrite the inequality in a more manageable form:

1. Identify when \((x - 9)^3 = 64\).

Taking the cube root of both sides, we have:
\[
x - 9 = \sqrt[3]{64}
\]
Since \(\sqrt[3]{64} = 4\), we can add 9 to both sides to solve for x:
\[
x - 9 = 4 \implies x = 4 + 9 = 13
\]

2. Next, we analyze the inequality \((x - 9)^3 < 64\):
The inequality implies:
\[
(x - 9)^3 < (4)^3
\]
Which gives us:
\[
(x - 9) < 4 \quad \text{or} \quad (x - 9) > -4
\]

3. Solve the first inequality:
\[
x - 9 < 4 \implies x < 13
\]

4. Solve the second inequality:
\[
x - 9 > -4 \implies x > 5
\]

5. Combine the results:
The solution for \((x - 9)^3 < 64\) is:
\[
5 < x < 13
\]

In interval notation, this solution is expressed as:
\[
(5, 13)
\]

Thus, the correct choice is:
**B. The solutions are the values of x in the interval (5, 13).**