Simplify the expression for the given values of \( x \). \( \left\lvert\, \frac{x}{3-x \mid-3}\right. \) if \( x \leq-3 \) Answer:

To simplify the expression \( \left\lvert\, \frac{x}{3-x \mid-3}\right. \) for \( x \leq -3 \), we first need to evaluate the denominator. Given that \( x \leq -3 \), we find that \( 3 - x \) is always positive because subtracting a negative number results in a larger positive value. Therefore, the expression simplifies as follows: 1. When \( x \leq -3 \), \( |3 - x| = 3 - x \). 2. Substitute this into the expression: \[ \left\lvert\, \frac{x}{3 - (-3)} \right\lvert = \left\lvert \frac{x}{6} \right\lvert \] 3. Since \( x \) is negative in this case, we can simplify to: \[ \left\lvert \frac{x}{6} \right\lvert = -\frac{x}{6} \] So the final simplified expression for \( x \leq -3 \) is \( -\frac{x}{6} \).