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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]
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]
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