### Video Transcript

Determine the solution set of the equation 31 minus six 𝑥 cubed is equal to negative 222𝑥 cubed minus 698 for all real values.

We can begin this question by adding 222𝑥 cubed and subtracting 31 from both sides of the equation. This gives us negative six 𝑥 cubed plus 222𝑥 cubed is equal to negative 698 minus 31. The left-hand side simplifies to 216𝑥 cubed. The right-hand side is equal to negative 729. Dividing both sides by 216 gives us 𝑥 cubed is equal to negative 729 over 216.

The opposite or inverse of cubing a number is cube rooting. 𝑥 is equal to the cube root of negative 729 over 216. Cube rooting any negative number gives a negative answer such that the cube root of negative 𝑥 is equal to negative the cube root of 𝑥. We also recall that when cube rooting a fraction, we can cube root the denominator and numerator separately. The cube root of 𝑎 over 𝑏 is equal to the cube root of 𝑎 over the cube root of 𝑏. Our value for 𝑥 is, therefore, equal to the negative of the cube root of 729 over the cube root of 216.

Six cubed is equal to 216. This means that the cube root of 216 is six. Likewise, nine cubed is 729. This means that the cube root of 729 is nine. 𝑥 is, therefore, equal to negative nine over six. As both of these numbers are divisible by three, the fraction simplifies to negative three over two.

The solution set of the equation is the single value negative three over two or negative 1.5.