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

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Answer

So, the solution to the inequality is $x < 18.49$ and $x > - 18.49$ .

Steps

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 $x < 18.49$ and $x > - 18.49$ .