\( \frac{1}{4}(x+4 x)+12=\frac{1}{2}(2-x)+10 \) \( 4 \times \quad \) A the equation, what is the value of \( x \) ? \( 4 \times \frac{4}{7} \) \( 4 \times \frac{7}{4} \) 4

To solve the equation \( \frac{1}{4}(x+4x)+12=\frac{1}{2}(2-x)+10 \), let’s first simplify both sides. Notice that \( \frac{1}{4}(x+4x) \) simplifies to \( \frac{5x}{4} \), leading to \( \frac{5x}{4} + 12 = \frac{1}{2}(2-x) + 10 \). Two can be expressed as \( 2 = \frac{4}{2} \), helping with the calculation. Now, multiply everything by 4 to eliminate the fractions, giving us \( 5x + 48 = 2(2-x) + 40 \). This simplifies to \( 5x + 48 = 4 - 2x + 40 \). Combining terms results in \( 5x + 48 = 44 - 2x \), so we add \( 2x \) to both sides and get \( 7x + 48 = 44 \). Subtracting 48 leads us to \( 7x = -4 \), and dividing by 7 gives \( x = -\frac{4}{7} \). Now, for the multiplication part: The expression \( 4 \times \frac{4}{7} \) calculates to \( \frac{16}{7} \), while \( 4 \times \frac{7}{4} \) simplifies to \( 7 \). So while both multiplications yield distinct values, keep in mind they showcase the beauty of fractions in different contexts! Let’s remember that \( 4 \) itself is just a simple integer - sometimes the simplest things can be the most impactful!