To solve the equation
\[ \left| \frac{1}{4}x + 7 \right| - 3 = 24, \]
we start by isolating the absolute value:
\[ \left| \frac{1}{4}x + 7 \right| = 24 + 3 = 27. \]
Next, we can split this into two cases based on the definition of absolute value.
Case 1:
\[ \frac{1}{4}x + 7 = 27. \]
To solve for \(x\), we subtract 7 from both sides:
\[ \frac{1}{4}x = 27 - 7 = 20. \]
Multiplying both sides by 4 gives:
\[ x = 20 \times 4 = 80. \]
Case 2:
\[ \frac{1}{4}x + 7 = -27. \]
Again, we subtract 7 from both sides:
\[ \frac{1}{4}x = -27 - 7 = -34. \]
Multiplying both sides by 4 gives:
\[ x = -34 \times 4 = -136. \]
Thus, the solutions to the equation are
\[ x = 80 \quad \text{and} \quad x = -136. \]
The correct answer is
\[ \boxed{b} \quad x = -136 \text{ and } x = 80. \]