Solve for x |2x-3/4|=5 6/7

$| 2 x - \frac{3}{4} | = 5 \frac{6}{7}$

Step 1

Convert $5 \frac{6}{7}$ to an improper fraction.

A mixed number is an addition of its whole and fractional parts.

$| 2 x - \frac{3}{4} | = 5 + \frac{6}{7}$

Add $5$ and $\frac{6}{7}$ .

To write $5$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$| 2 x - \frac{3}{4} | = 5 \cdot \frac{7}{7} + \frac{6}{7}$

Combine $5$ and $\frac{7}{7}$ .

$| 2 x - \frac{3}{4} | = \frac{5 \cdot 7}{7} + \frac{6}{7}$

Combine the numerators over the common denominator.

$| 2 x - \frac{3}{4} | = \frac{5 \cdot 7 + 6}{7}$

Simplify the numerator.

Multiply $5$ by $7$ .

$| 2 x - \frac{3}{4} | = \frac{35 + 6}{7}$

Add $35$ and $6$ .

$| 2 x - \frac{3}{4} | = \frac{41}{7}$

Step 2

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

$2 x - \frac{3}{4} = \pm \frac{41}{7}$

Step 3

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.

$2 x - \frac{3}{4} = \frac{41}{7}$

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

Add $\frac{3}{4}$ to both sides of the equation.

$2 x = \frac{41}{7} + \frac{3}{4}$

To write $\frac{41}{7}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 x = \frac{41}{7} \cdot \frac{4}{4} + \frac{3}{4}$

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$2 x = \frac{41}{7} \cdot \frac{4}{4} + \frac{3}{4} \cdot \frac{7}{7}$

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{41}{7}$ by $\frac{4}{4}$ .

$2 x = \frac{41 \cdot 4}{7 \cdot 4} + \frac{3}{4} \cdot \frac{7}{7}$

Multiply $7$ by $4$ .

$2 x = \frac{41 \cdot 4}{28} + \frac{3}{4} \cdot \frac{7}{7}$

Multiply $\frac{3}{4}$ by $\frac{7}{7}$ .

$2 x = \frac{41 \cdot 4}{28} + \frac{3 \cdot 7}{4 \cdot 7}$

Multiply $4$ by $7$ .

$2 x = \frac{41 \cdot 4}{28} + \frac{3 \cdot 7}{28}$

Combine the numerators over the common denominator.

$2 x = \frac{41 \cdot 4 + 3 \cdot 7}{28}$

Simplify the numerator.

Multiply $41$ by $4$ .

$2 x = \frac{164 + 3 \cdot 7}{28}$

Multiply $3$ by $7$ .

$2 x = \frac{164 + 21}{28}$

Add $164$ and $21$ .

$2 x = \frac{185}{28}$

Divide each term in $2 x = \frac{185}{28}$ by $2$ and simplify.

Divide each term in $2 x = \frac{185}{28}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{185}{28}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{185}{28}}{2}$

Divide $x$ by $1$ .

$x = \frac{\frac{185}{28}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{185}{28} \cdot \frac{1}{2}$

Multiply $\frac{185}{28} \cdot \frac{1}{2}$ .

Multiply $\frac{185}{28}$ by $\frac{1}{2}$ .

$x = \frac{185}{28 \cdot 2}$

Multiply $28$ by $2$ .

$x = \frac{185}{56}$

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

$2 x - \frac{3}{4} = - \frac{41}{7}$

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

Add $\frac{3}{4}$ to both sides of the equation.

$2 x = - \frac{41}{7} + \frac{3}{4}$

To write $- \frac{41}{7}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 x = - \frac{41}{7} \cdot \frac{4}{4} + \frac{3}{4}$

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$2 x = - \frac{41}{7} \cdot \frac{4}{4} + \frac{3}{4} \cdot \frac{7}{7}$

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{41}{7}$ by $\frac{4}{4}$ .

$2 x = - \frac{41 \cdot 4}{7 \cdot 4} + \frac{3}{4} \cdot \frac{7}{7}$

Multiply $7$ by $4$ .

$2 x = - \frac{41 \cdot 4}{28} + \frac{3}{4} \cdot \frac{7}{7}$

Multiply $\frac{3}{4}$ by $\frac{7}{7}$ .

$2 x = - \frac{41 \cdot 4}{28} + \frac{3 \cdot 7}{4 \cdot 7}$

Multiply $4$ by $7$ .

$2 x = - \frac{41 \cdot 4}{28} + \frac{3 \cdot 7}{28}$

Combine the numerators over the common denominator.

$2 x = \frac{- 41 \cdot 4 + 3 \cdot 7}{28}$

Simplify the numerator.

Multiply $- 41$ by $4$ .

$2 x = \frac{- 164 + 3 \cdot 7}{28}$

Multiply $3$ by $7$ .

$2 x = \frac{- 164 + 21}{28}$

Add $- 164$ and $21$ .

$2 x = \frac{- 143}{28}$

Move the negative in front of the fraction.

$2 x = - \frac{143}{28}$

Divide each term in $2 x = - \frac{143}{28}$ by $2$ and simplify.

Divide each term in $2 x = - \frac{143}{28}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{143}{28}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{143}{28}}{2}$

Divide $x$ by $1$ .

$x = \frac{- \frac{143}{28}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{143}{28} \cdot \frac{1}{2}$

Multiply $- \frac{143}{28} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{143}{28}$ .

$x = - \frac{143}{2 \cdot 28}$

Multiply $2$ by $28$ .

$x = - \frac{143}{56}$

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

$x = \frac{185}{56}, - \frac{143}{56}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{185}{56}, - \frac{143}{56}$

Decimal Form:

$x = 3.30357142 \dots, - 2.55357142 \dots$

Mixed Number Form:

$x = 3 \frac{17}{56}, - 2 \frac{31}{56}$

Solve for x |2x-3/4|=5 6/7

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