To solve the equation
\[ \frac{4}{7}\left(\frac{21}{8}x + \frac{1}{2}\right) = -2\left(\frac{1}{7} - \frac{5}{28}x\right), \]
we start by simplifying both sides.
Step 1: Simplifying the left side
Distributing \(\frac{4}{7}\):
\[ \frac{4}{7} \cdot \frac{21}{8}x + \frac{4}{7} \cdot \frac{1}{2}. \]
Calculating \(\frac{4}{7} \cdot \frac{21}{8}x\):
\[ \frac{4 \cdot 21}{7 \cdot 8}x = \frac{84}{56}x = \frac{3}{2}x. \]
Next, calculating \(\frac{4}{7} \cdot \frac{1}{2}\):
\[ \frac{4 \cdot 1}{7 \cdot 2} = \frac{4}{14} = \frac{2}{7}. \]
So the left side simplifies to:
\[ \frac{3}{2}x + \frac{2}{7}. \]
Step 2: Simplifying the right side
Distributing \(-2\):
\[ -2 \cdot \frac{1}{7} + (-2) \cdot \left(- \frac{5}{28}x\right). \]
Calculating \(-2 \cdot \frac{1}{7}\):
\[ -\frac{2}{7}. \]
Calculating \(-2 \cdot \left(-\frac{5}{28}x\right)\):
\[ \frac{10}{28}x = \frac{5}{14}x. \]
So the right side simplifies to:
\[ -\frac{2}{7} + \frac{5}{14}x. \]
Step 3: Setting the simplified forms equal
Now we have
\[ \frac{3}{2}x + \frac{2}{7} = -\frac{2}{7} + \frac{5}{14}x. \]
Step 4: Combining like terms
To eliminate the fractions, we can multiply through by the least common multiple of the denominators, which is 14:
\[ 14\left(\frac{3}{2}x + \frac{2}{7}\right) = 14\left(-\frac{2}{7} + \frac{5}{14}x\right). \]
Calculating each side:
\[ 14 \cdot \frac{3}{2}x + 14 \cdot \frac{2}{7} = 21x + 4, \] \[ 14 \cdot -\frac{2}{7} + 14 \cdot \frac{5}{14}x = -4 + 5x. \]
So we have:
\[ 21x + 4 = -4 + 5x. \]
Step 5: Moving terms involving \(x\) to one side
Rearranging gives:
\[ 21x - 5x = -4 - 4. \]
This simplifies to:
\[ 16x = -8. \]
Step 6: Solving for \(x\)
Dividing both sides by 16:
\[ x = -\frac{8}{16} = -\frac{1}{2}. \]
Decimal Form:
Thus, in decimal form, the solution is:
\[ x = -0.5. \]
So the final answer is:
\[ \boxed{-0.5}. \]
1 answer