To find the value of [tex]\( x \)[/tex] in the equation [tex]\(\frac{4}{7}\left(\frac{21}{8} x+\frac{1}{2}\right)=-2\left(\frac{1}{7}-\frac{5}{28} x\right)\)[/tex], follow these steps:
1. Distribute the fractions inside the parentheses:
[tex]\[
\frac{4}{7} \left(\frac{21}{8} x + \frac{1}{2}\right)
\][/tex]
Distribute [tex]\(\frac{4}{7}\)[/tex]:
[tex]\[
= \frac{4}{7} \cdot \frac{21}{8} x + \frac{4}{7} \cdot \frac{1}{2}
\][/tex]
Simplify each term:
[tex]\[
= \left(\frac{4 \times 21}{7 \times 8}\right) x + \left(\frac{4 \times 1}{7 \times 2}\right)
\][/tex]
[tex]\[
= \left(\frac{84}{56}\right) x + \left(\frac{4}{14}\right)
\][/tex]
Simplify the fractions:
[tex]\[
= \left(\frac{3}{2}\right) x + \left(\frac{2}{7}\right)
\][/tex]
2. Distribute on the right-hand side:
[tex]\[
-2 \left(\frac{1}{7} - \frac{5}{28} x\right)
\][/tex]
Distribute [tex]\(-2\)[/tex]:
[tex]\[
= -2 \cdot \frac{1}{7} + (-2) \cdot \left(-\frac{5}{28} x\right)
\][/tex]
Simplify each term:
[tex]\[
= -\frac{2}{7} + \left(\frac{10}{28}\right) x
\][/tex]
[tex]\[
= -\frac{2}{7} + \frac{5}{14} x
\][/tex]
3. Combine the simplified expressions:
Now we have:
[tex]\[
\frac{3}{2} x + \frac{2}{7} = -\frac{2}{7} + \frac{5}{14} x
\][/tex]
4. Move all terms involving [tex]\( x \)[/tex] to one side and constants to the other:
Subtract [tex]\(\frac{5}{14} x\)[/tex] from both sides:
[tex]\[
\frac{3}{2} x - \frac{5}{14} x + \frac{2}{7} = -\frac{2}{7}
\][/tex]
Convert [tex]\(\frac{3}{2}\)[/tex] and [tex]\(\frac{5}{14}\)[/tex] to a common base:
[tex]\[
\frac{21}{14} x - \frac{5}{14} x + \frac{2}{7} = -\frac{2}{7}
\][/tex]
Combine the fractions:
[tex]\[
\left(\frac{21 - 5}{14}\right) x + \frac{2}{7} = -\frac{2}{7}
\][/tex]
Simplify:
[tex]\[
\frac{16}{14} x + \frac{2}{7} = -\frac{2}{7}
\][/tex]
Convert [tex]\(\frac{16}{14} x\)[/tex] to simplest form:
[tex]\[
\frac{8}{7} x + \frac{2}{7} = -\frac{2}{7}
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Subtract [tex]\(\frac{2}{7}\)[/tex] from both sides:
[tex]\[
\frac{8}{7} x = -\frac{2}{7} - \frac{2}{7}
\][/tex]
Simplify:
[tex]\[
\frac{8}{7} x = -\frac{4}{7}
\][/tex]
Multiply both sides by [tex]\(\frac{7}{8}\)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{4}{7} \cdot \frac{7}{8}
\][/tex]
[tex]\[
x = -\frac{4}{8}
\][/tex]
[tex]\[
x = -\frac{1}{2}
\][/tex]
In decimal form, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = -0.5
\][/tex]
