Video Transcript
Find the solution set of the inequality cube root of 64 is less than or equal to 𝑥 minus nine is less than or equal to the square root of 81 in the set of real numbers. Give your answer in interval notation.
We first note that because four cubed is 64, the cube root of 64 is four. And because nine squared is 81, the square root of 81 is nine. This allows us to simplify the given inequality to four is less than or equal to 𝑥 minus nine is less than or equal to nine.
Since the variable 𝑥 only appears in the middle of the compound inequality, we can solve this inequality by isolating 𝑥 in the middle. To do this, we add nine to all parts of the inequality to get the equivalent inequality. 13 is less than or equal to 𝑥 is less than or equal to 18. Therefore, the values of 𝑥 that satisfy the compound inequality are all of the real values between and including 13 and 18.
We can sketch this set on a number line by connecting a closed circle at 13 to a closed circle at 18. Finally, we need to write this in interval notation. We use closed brackets on both the left and the right to show that 13 and 18 are included in the solution set and satisfy the inequality.
Therefore, the values that satisfy the inequality are the real numbers in the closed interval from 13 to 18.
