1. **State the problem:** Solve the equation $$2|4 - x| + 3|4 - x| = 25$$ for the positive value of $x$. 2. **Combine like terms:** Since both terms involve $|4 - x|$, add the coefficients: $$2|4 - x| + 3|4 - x| = (2 + 3)|4 - x| = 5|4 - x|$$ 3. **Rewrite the equation:** $$5|4 - x| = 25$$ 4. **Isolate the absolute value:** $$|4 - x| = \frac{25}{5}$$ $$|4 - x| = 5$$ 5. **Solve the absolute value equation:** The definition of absolute value gives two cases: - Case 1: $$4 - x = 5$$ - Case 2: $$4 - x = -5$$ 6. **Solve Case 1:** $$4 - x = 5$$ $$-x = 5 - 4$$ $$-x = 1$$ $$x = \cancel{-1} \times \cancel{-1} = 1$$ 7. **Solve Case 2:** $$4 - x = -5$$ $$-x = -5 - 4$$ $$-x = -9$$ $$x = \cancel{-9} \times \cancel{-1} = 9$$ 8. **Identify the positive solution:** Both $x=1$ and $x=9$ are positive, but the problem asks for the positive solution, so both qualify. Usually, if only one positive solution is requested, it is the larger or the one that fits the context. Here, both are positive, so both are solutions. **Final answer:** The positive solutions are $$x = 1$$ and $$x = 9$$. If only one positive solution is requested, clarify the context; otherwise, both are valid.