for what values of the variable does each of the following expressions make sense?

sqaure root of -(6-x)

1 answer

The expression \( \sqrt{-(6 - x)} \) involves a square root, and for the square root to be defined in the real number system, the expression inside the square root must be non-negative. Therefore, we need to ensure that:

\[ -(6 - x) \geq 0 \]

This can be rearranged as follows:

  1. Multiply both sides by -1 (remember to reverse the inequality sign): \[ 6 - x \leq 0 \]

  2. Rearranging gives: \[ 6 \leq x \] or \[ x \geq 6 \]

Thus, the expression \( \sqrt{-(6 - x)} \) is defined for values of \( x \) that are greater than or equal to 6.

In interval notation, the solution is: \[ [6, \infty) \]