Final answer:
To find the positive solution to the equation 2|4-x|+3|4-x|=25, combine like terms to get 5|4-x|=25, then isolate the absolute value to |4-x|=5. Solve for x, considering the absolute value property, which leads to the positive solution x=9.
Step-by-step explanation:
The question asks us to solve the equation 2|4-x|+3|4-x|=25 for the positive solution of x. To simplify the problem, we first combine the like terms on the left side of the equation. Since both terms involve the absolute value of (4-x), we can add them together to get 5|4-x|. This yields:
5|4-x| = 25
Now, we divide both sides of the equation by 5 to isolate the absolute value:
|4-x| = 5
Next, we consider the nature of absolute value; there can be two solutions for x: one where (4-x) is positive and equal to 5, and one where (4-x) is negative and equal to -5.
For the first case:
- 4 - x = 5
- -x = 5 - 4
- -x = 1
- x = -1 (negative solution)
For the second case:
- 4 - x = -5
- - x = -5 - 4
- -x = -9
- x = 9 (positive solution)
Since we are looking for the positive solution, the answer is x = 9.