Final answer:
To solve |7-x|+4=2x+5, one must consider the two cases of the absolute value expression. Only one value of x, specifically x = 2, satisfies the original equation. The value x = -8 does not, as it fails to meet the required conditions.
Step-by-step explanation:
To solve the equation |7-x|+4=2x+5, we need to consider both cases for the absolute value, one where 7-x is positive, and one where 7-x is negative. Starting with the first case, for 7 - x ≥ 0, the equation simplifies to:
7 - x + 4 = 2x + 5
Rearrange and solve for x:
7 + 4 - 5 = 2x + x
6 = 3x
x = 2
For the second case, for 7 - x < 0, we consider the equation:
-(7 - x) + 4 = 2x + 5
-7 + x + 4 = 2x + 5
-3 = x + 5
x = -8
However, if we substitute x = -8 back into the original equation, we find that it does not satisfy the condition 7 - x < 0. Therefore, x = -8 is not a solution to the equation. The only solution is x = 2, which we can check by substituting back into the original equation:
|7 - 2| + 4 = 2 × 2 + 5
|5| + 4 = 4 + 5
5 + 4 = 9
9 = 9
The solution x = 2 validates the equation, hence it is the correct answer.