1. **State the problem:** Solve the equation $$11 - 3 \sqrt{28} x = 52$$ for $x$. 2. **Isolate the term with $x$:** Subtract 11 from both sides: $$11 - 3 \sqrt{28} x - 11 = 52 - 11$$ $$-3 \sqrt{28} x = 41$$ 3. **Divide both sides by $-3 \sqrt{28}$ to solve for $x$:** $$x = \frac{41}{-3 \sqrt{28}}$$ Show the cancellation step: $$x = \frac{\cancel{41}}{\cancel{-3} \sqrt{28}} \times \frac{\cancel{1}}{\cancel{1}} = -\frac{41}{3 \sqrt{28}}$$ 4. **Simplify $\sqrt{28}$:** $$\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2 \sqrt{7}$$ 5. **Substitute back and simplify:** $$x = -\frac{41}{3 \times 2 \sqrt{7}} = -\frac{41}{6 \sqrt{7}}$$ 6. **Rationalize the denominator:** Multiply numerator and denominator by $\sqrt{7}$: $$x = -\frac{41 \sqrt{7}}{6 \sqrt{7} \times \sqrt{7}} = -\frac{41 \sqrt{7}}{6 \times 7} = -\frac{41 \sqrt{7}}{42}$$ **Final answer:** $$x = -\frac{41 \sqrt{7}}{42}$$