Solve for x |3/7x+6|-1/2=8

$| \frac{3}{7} x + 6 | - \frac{1}{2} = 8$

Move all terms not containing $| \frac{3}{7} x + 6 |$ to the right side of the equation.

Add $\frac{1}{2}$ to both sides of the equation.

$| \frac{3}{7} x + 6 | = 8 + \frac{1}{2}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$| \frac{3}{7} x + 6 | = 8 \cdot \frac{2}{2} + \frac{1}{2}$

Combine $8$ and $\frac{2}{2}$ .

$| \frac{3}{7} x + 6 | = \frac{8 \cdot 2}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$| \frac{3}{7} x + 6 | = \frac{8 \cdot 2 + 1}{2}$

Simplify the numerator.

Multiply $8$ by $2$ .

$| \frac{3}{7} x + 6 | = \frac{16 + 1}{2}$

Add $16$ and $1$ .

$| \frac{3}{7} x + 6 | = \frac{17}{2}$

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$\frac{3}{7} x + 6 = \pm \frac{17}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$\frac{3}{7} x + 6 = \frac{17}{2}$

Combine $\frac{3}{7}$ and $x$ .

$\frac{3 x}{7} + 6 = \frac{17}{2}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $6$ from both sides of the equation.

$\frac{3 x}{7} = \frac{17}{2} - 6$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{7} = \frac{17}{2} - 6 \cdot \frac{2}{2}$

Combine $- 6$ and $\frac{2}{2}$ .

$\frac{3 x}{7} = \frac{17}{2} + \frac{- 6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$\frac{3 x}{7} = \frac{17 - 6 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 6$ by $2$ .

$\frac{3 x}{7} = \frac{17 - 12}{2}$

Subtract $12$ from $17$ .

$\frac{3 x}{7} = \frac{5}{2}$

Multiply both sides of the equation by $\frac{7}{3}$ .

$\frac{7}{3} \cdot \frac{3 x}{7} = \frac{7}{3} \cdot \frac{5}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{7}{3} \cdot \frac{3 x}{7}$ .

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7}{3} \cdot \frac{3 x}{7} = \frac{7}{3} \cdot \frac{5}{2}$

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{7}{3} \cdot \frac{5}{2}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{7}{3} \cdot \frac{5}{2}$

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{7}{3} \cdot \frac{5}{2}$

Rewrite the expression.

$x = \frac{7}{3} \cdot \frac{5}{2}$

Simplify the right side.

Multiply $\frac{7}{3} \cdot \frac{5}{2}$ .

Multiply $\frac{7}{3}$ by $\frac{5}{2}$ .

$x = \frac{7 \cdot 5}{3 \cdot 2}$

Multiply $7$ by $5$ .

$x = \frac{35}{3 \cdot 2}$

Multiply $3$ by $2$ .

$x = \frac{35}{6}$

Next, use the negative value of the $\pm$ to find the second solution.

$\frac{3}{7} x + 6 = - \frac{17}{2}$

Combine $\frac{3}{7}$ and $x$ .

$\frac{3 x}{7} + 6 = - \frac{17}{2}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $6$ from both sides of the equation.

$\frac{3 x}{7} = - \frac{17}{2} - 6$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{7} = - \frac{17}{2} - 6 \cdot \frac{2}{2}$

Combine $- 6$ and $\frac{2}{2}$ .

$\frac{3 x}{7} = - \frac{17}{2} + \frac{- 6 \cdot 2}{2}$

Combine the numerators over the common denominator.

$\frac{3 x}{7} = \frac{- 17 - 6 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 6$ by $2$ .

$\frac{3 x}{7} = \frac{- 17 - 12}{2}$

Subtract $12$ from $- 17$ .

$\frac{3 x}{7} = \frac{- 29}{2}$

Move the negative in front of the fraction.

$\frac{3 x}{7} = - \frac{29}{2}$

Multiply both sides of the equation by $\frac{7}{3}$ .

$\frac{7}{3} \cdot \frac{3 x}{7} = \frac{7}{3} (- \frac{29}{2})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{7}{3} \cdot \frac{3 x}{7}$ .

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7}{3} \cdot \frac{3 x}{7} = \frac{7}{3} (- \frac{29}{2})$

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{7}{3} (- \frac{29}{2})$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{7}{3} (- \frac{29}{2})$

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{7}{3} (- \frac{29}{2})$

Rewrite the expression.

$x = \frac{7}{3} (- \frac{29}{2})$

Simplify the right side.

Multiply $\frac{7}{3} (- \frac{29}{2})$ .

Multiply $\frac{7}{3}$ by $\frac{29}{2}$ .

$x = - \frac{7 \cdot 29}{3 \cdot 2}$

Multiply $7$ by $29$ .

$x = - \frac{203}{3 \cdot 2}$

Multiply $3$ by $2$ .

$x = - \frac{203}{6}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{35}{6}, - \frac{203}{6}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{35}{6}, - \frac{203}{6}$

Decimal Form:

$x = 5.8\overline{3}, - 33.8\overline{3}$

Mixed Number Form:

$x = 5 \frac{5}{6}, - 33 \frac{5}{6}$

Solve for x |3/7x+6|-1/2=8

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