For what values of x is the following expression a real number? \( \frac{3}{\sqrt{21-3 x}} \)

To determine for which values of \( x \) the expression \[ \frac{3}{\sqrt{21 - 3x}} \] is a real number, we need to ensure two main conditions: 1. **The expression inside the square root must be non-negative.** The square root function is only defined for non-negative numbers in the real number system. Therefore, we require: \[ 21 - 3x \geq 0 \] 2. **The denominator cannot be zero.** If the denominator were zero, the expression would be undefined. Hence: \[ \sqrt{21 - 3x} \neq 0 \quad \Rightarrow \quad 21 - 3x > 0 \] Let's solve the inequality: \[ 21 - 3x > 0 \] Subtract 21 from both sides: \[ -3x > -21 \] Now, divide both sides by \(-3\). **Remember** that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign: \[ x < 7 \] **Conclusion:** The expression \(\frac{3}{\sqrt{21 - 3x}}\) is a real number for all real values of \( x \) that are **less than 7**. **Final Answer:** All real numbers below seven. In other words, the expression is real for every real x with x  <  7.