(d) \( |x+7|=|x+12| \)

To solve the equation \( |x+7|=|x+12| \), we can break it down into two cases based on the definition of absolute value. **Case 1:** \( x + 7 = x + 12 \) This simplifies to \( 7 = 12 \), which is not true. **Case 2:** \( x + 7 = -(x + 12) \) This simplifies to \( x + 7 = -x - 12 \), leading to \( 2x = -19 \), or \( x = -\frac{19}{2} \). Now, let's check our solution by substituting it back into the original equation. Calculating \( |-\frac{19}{2} + 7| = |-\frac{19}{2} + 12| \): On the left: \( |-\frac{19}{2} + \frac{14}{2}| = |\frac{-5}{2}| = \frac{5}{2} \), On the right: \( |-\frac{19}{2} + \frac{24}{2}| = |\frac{5}{2}| =\frac{5}{2} \). Since both sides are equal, \( x = -\frac{19}{2} \) is indeed the solution!