To solve the equation \( \frac{1}{4}(4x + 6) = -5x + 24.7 \), we can start by simplifying the left side.
First, distribute \( \frac{1}{4} \):
\[ \frac{1}{4} \cdot 4x + \frac{1}{4} \cdot 6 = x + \frac{3}{2} \]
So the equation becomes:
\[ x + \frac{3}{2} = -5x + 24.7 \]
Next, we want to get all terms involving \( x \) on one side of the equation and constant terms on the other side. Add \( 5x \) to both sides:
\[ x + 5x + \frac{3}{2} = 24.7 \]
This simplifies to:
\[ 6x + \frac{3}{2} = 24.7 \]
Now, subtract \( \frac{3}{2} \) from both sides. First, convert \( 24.7 \) into a fraction:
\[ 24.7 = \frac{247}{10} \]
Now we need a common denominator to subtract \( \frac{3}{2} \):
Convert \( \frac{3}{2} \) to tenths:
\[ \frac{3}{2} = \frac{15}{10} \]
Now we can subtract:
\[ 6x = \frac{247}{10} - \frac{15}{10} = \frac{232}{10} \]
This simplifies to:
\[ 6x = 23.2 \]
Now, divide both sides by 6:
\[ x = \frac{23.2}{6} \approx 3.8667 \]
Thus, the solution to the equation is:
\[ x \approx 3.8667 \]
If you prefer the exact fraction form, we can simplify:
\[ x = \frac{116}{30} \text{ (which simplifies to } \frac{58}{15}\text{)} \]
So the exact solution is:
\[ x = \frac{58}{15} \]