Use algebraic​ and/or graphical methods to solve the inequality.

Question

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).