Finite Math Examples

$\frac{x}{2 x - 1} - 3 = \frac{1}{4}$

Step 1

Subtract $\frac{1}{4}$ from both sides of the equation.

$\frac{x}{2 x - 1} - 3 - \frac{1}{4} = 0$

Step 2

Simplify $\frac{x}{2 x - 1} - 3 - \frac{1}{4}$ .

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Step 2.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{x}{2 x - 1} - 3 \cdot \frac{2 x - 1}{2 x - 1} - \frac{1}{4} = 0$

Step 2.2

Combine $- 3$ and $\frac{2 x - 1}{2 x - 1}$ .

$\frac{x}{2 x - 1} + \frac{- 3 (2 x - 1)}{2 x - 1} - \frac{1}{4} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x - 3 (2 x - 1)}{2 x - 1} - \frac{1}{4} = 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{x - 3 (2 x) - 3 \cdot - 1}{2 x - 1} - \frac{1}{4} = 0$

Step 2.4.2

Multiply $2$ by $- 3$ .

$\frac{x - 6 x - 3 \cdot - 1}{2 x - 1} - \frac{1}{4} = 0$

Step 2.4.3

Multiply $- 3$ by $- 1$ .

$\frac{x - 6 x + 3}{2 x - 1} - \frac{1}{4} = 0$

Step 2.4.4

Subtract $6 x$ from $x$ .

$\frac{- 5 x + 3}{2 x - 1} - \frac{1}{4} = 0$

Step 2.5

To write $\frac{- 5 x + 3}{2 x - 1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{- 5 x + 3}{2 x - 1} \cdot \frac{4}{4} - \frac{1}{4} = 0$

Step 2.6

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{- 5 x + 3}{2 x - 1} \cdot \frac{4}{4} - \frac{1}{4} \cdot \frac{2 x - 1}{2 x - 1} = 0$

Step 2.7

Write each expression with a common denominator of $(2 x - 1) \cdot 4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.7.1

Multiply $\frac{- 5 x + 3}{2 x - 1}$ by $\frac{4}{4}$ .

$\frac{(- 5 x + 3) \cdot 4}{(2 x - 1) \cdot 4} - \frac{1}{4} \cdot \frac{2 x - 1}{2 x - 1} = 0$

Step 2.7.2

Multiply $\frac{1}{4}$ by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{(- 5 x + 3) \cdot 4}{(2 x - 1) \cdot 4} - \frac{2 x - 1}{4 (2 x - 1)} = 0$

Step 2.7.3

Reorder the factors of $(2 x - 1) \cdot 4$ .

$\frac{(- 5 x + 3) \cdot 4}{4 (2 x - 1)} - \frac{2 x - 1}{4 (2 x - 1)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{(- 5 x + 3) \cdot 4 - (2 x - 1)}{4 (2 x - 1)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{- 5 x \cdot 4 + 3 \cdot 4 - (2 x - 1)}{4 (2 x - 1)} = 0$

Step 2.9.2

Multiply $4$ by $- 5$ .

$\frac{- 20 x + 3 \cdot 4 - (2 x - 1)}{4 (2 x - 1)} = 0$

Step 2.9.3

Multiply $3$ by $4$ .

$\frac{- 20 x + 12 - (2 x - 1)}{4 (2 x - 1)} = 0$

Step 2.9.4

Apply the distributive property.

$\frac{- 20 x + 12 - (2 x) - - 1}{4 (2 x - 1)} = 0$

Step 2.9.5

Multiply $2$ by $- 1$ .

$\frac{- 20 x + 12 - 2 x - - 1}{4 (2 x - 1)} = 0$

Step 2.9.6

Multiply $- 1$ by $- 1$ .

$\frac{- 20 x + 12 - 2 x + 1}{4 (2 x - 1)} = 0$

Step 2.9.7

Subtract $2 x$ from $- 20 x$ .

$\frac{- 22 x + 12 + 1}{4 (2 x - 1)} = 0$

Step 2.9.8

Add $12$ and $1$ .

$\frac{- 22 x + 13}{4 (2 x - 1)} = 0$

Step 2.10

Factor $- 1$ out of $- 22 x$ .

$\frac{- (22 x) + 13}{4 (2 x - 1)} = 0$

Step 2.11

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (22 x) - 1 (- 13)}{4 (2 x - 1)} = 0$

Step 2.12

Factor $- 1$ out of $- (22 x) - 1 (- 13)$ .

$\frac{- (22 x - 13)}{4 (2 x - 1)} = 0$

Step 2.13

Rewrite $- (22 x - 13)$ as $- 1 (22 x - 13)$ .

$\frac{- 1 (22 x - 13)}{4 (2 x - 1)} = 0$

Step 2.14

Move the negative in front of the fraction.

$- \frac{22 x - 13}{4 (2 x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$22 x - 13 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Add $13$ to both sides of the equation.

$22 x = 13$

Step 4.2

Divide each term in $22 x = 13$ by $22$ and simplify.

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Step 4.2.1

Divide each term in $22 x = 13$ by $22$ .

$\frac{22 x}{22} = \frac{13}{22}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $22$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{22 x}{22} = \frac{13}{22}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{22}$