Question

What is the solution for this inequality? 8x⁵-32 A. X=-4 B. X=4 c. X²-4 D x = 4

Answer 1

The solution for this inequality is:

B. X = 4

Step-by-step explanation:

The given inequality is
`\(8x^5 - 32 < 0\).` To find the solution, we can first factor out the common factor of 8:

`\[8(x^5 - 4) < 0\]`

Now, we set each factor equal to zero to find critical points:

`\[x^5 - 4 = 0\]`

Solving for
`\(x\), we get \(x = 4\)`. This critical point divides the real number line into two intervals:
`\((-\infty, 4)\) and \((4, \infty)\).`

To determine the sign of
`\(8(x^5 - 4)\)` in each interval, we choose test points. Taking a test point from the interval
`\((-\infty, 4)\), say \(x = 0\)`, we substitute it into the inequality:

`\[8(0^5 - 4) < 0\]`

This simplifies to
`\(8(-4) < 0\),` which is true. Therefore, the solution lies in the interval
`\((-\infty, 4)\).`

Now, taking a test point from the interval
`\((4, \infty)\), say \(x = 5\)`, we substitute it into the inequality:

`\[8(5^5 - 4) < 0\]`

This simplifies to
`\(8(311) > 0\),` which is false. Therefore, the solution does not lie in the interval
`\((4, \infty)\).`

Combining both intervals, the solution to
`\(8x^5 - 32 < 0\) is \(x \in (-\infty, 4)\),` and the correct answer is
`\(B. X = 4\).`