Question

What is the value of x in the equation StartFraction 4 Over 7 EndFraction (StartFraction 21 Over 8 EndFraction x + one-half) = negative 2 (StartFraction 1 Over 7 EndFraction minus StartFraction 5 Over 28 EndFraction x)? Write your solution in decimal form.

Answer 1

To solve the equation

\[
\frac{4}{7} \left(\frac{21}{8} x + \frac{1}{2}\right) = -2 \left(\frac{1}{7} - \frac{5}{28} x\right),
\]

let's start by simplifying both sides step by step.

1. **Left Side:**

The left side is

\[
\frac{4}{7} \left(\frac{21}{8} x + \frac{1}{2}\right).
\]

Distributing \(\frac{4}{7}\):

\[
\frac{4}{7} \cdot \frac{21}{8} x + \frac{4}{7} \cdot \frac{1}{2}.
\]

Let's calculate each term separately:

- For the \(x\) term:
\[
\frac{4 \cdot 21}{7 \cdot 8} x = \frac{84}{56} x = \frac{3}{2} x.
\]

- For the constant term:
\[
\frac{4 \cdot 1}{7 \cdot 2} = \frac{4}{14} = \frac{2}{7}.
\]

Thus, the left side simplifies to:

\[
\frac{3}{2} x + \frac{2}{7}.
\]

2. **Right Side:**

The right side is

\[
-2 \left(\frac{1}{7} - \frac{5}{28} x\right).
\]

Distributing \(-2\):

\[
-2 \cdot \frac{1}{7} + 2 \cdot \frac{5}{28} x.
\]

Calculating each term:

- For the constant term:
\[
-\frac{2}{7}.
\]

- For the \(x\) term:
\[
\frac{10}{28} x = \frac{5}{14} x.
\]

Thus, the right side simplifies to:

\[
-\frac{2}{7} + \frac{5}{14} x.
\]

3. **Setting the two sides equal:**

Now we have the equation:

\[
\frac{3}{2} x + \frac{2}{7} = -\frac{2}{7} + \frac{5}{14} x.
\]

First, eliminate the fractions by finding a common denominator, which is 14. Multiply each term by 14:

\[
14 \left(\frac{3}{2} x\right) + 14 \left(\frac{2}{7}\right) = 14 \left(-\frac{2}{7}\right) + 14 \left(\frac{5}{14} x\right).
\]

Simplifying each term results in:

\[
21x + 4 = -4 + 5x.
\]

4. **Rearranging the equation:**

Now, let's move all \(x\) terms to one side and constants to the other:

\[
21x - 5x = -4 - 4,
\]
leading to

\[
16x = -8.
\]

5. **Solving for \(x\):**

Now divide both sides by 16:

\[
x = -\frac{8}{16} = -\frac{1}{2}.
\]

6. **Writing in decimal form:**

Converting \(-\frac{1}{2}\) to decimal gives:

\[
x = -0.5.
\]

Thus, the final answer is

\[
\boxed{-0.5}.
\]