Solve the following inequality.
\[
\sqrt[3]{-x^{2}-1}> -7
\]
Check for extraneous solutions. Round your answers to two decimal places.
$x< $ or $x> $
\[
< x<
\]
Answer
So, the solution to the inequality is \(x < \boxed{18.49}\) and \(x > \boxed{-18.49}\).
Step 1 ：First, cube both sides of the inequality to remove the cube root. This gives us \(-x^{2}-1 > -343\).
Step 2 ：Next, isolate \(x^{2}\) by adding 1 to both sides and multiplying by -1. This gives us \(x^{2} < 342\).
Step 3 ：Finally, solve for \(x\) by taking the square root of both sides. This gives us \(x < \sqrt{342}\) and \(x > -\sqrt{342}\).
Step 4 ：Calculating the square root of 342 gives approximately 18.49.
Step 5 ：So, the solution to the inequality is \(x < \boxed{18.49}\) and \(x > \boxed{-18.49}\).
