Simplify (x-4)/(2x)+(3x)/(2x-1)

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Expand $(x - 4) (2 x - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 + 3 x (2 x)}{2 x (2 x - 1)}$

Move $- 1$ to the left of $x$ .

$\frac{2 x^{2} - 1 \cdot x - 4 (2 x) - 4 \cdot - 1 + 3 x (2 x)}{2 x (2 x - 1)}$

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x^{2} - x - 4 (2 x) - 4 \cdot - 1 + 3 x (2 x)}{2 x (2 x - 1)}$

Multiply $2$ by $- 4$ .

$\frac{2 x^{2} - x - 8 x - 4 \cdot - 1 + 3 x (2 x)}{2 x (2 x - 1)}$

Multiply $- 4$ by $- 1$ .

$\frac{2 x^{2} - x - 8 x + 4 + 3 x (2 x)}{2 x (2 x - 1)}$

Subtract $8 x$ from $- x$ .

$\frac{2 x^{2} - 9 x + 4 + 3 x (2 x)}{2 x (2 x - 1)}$

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} - 9 x + 4 + 3 \cdot 2 x \cdot x}{2 x (2 x - 1)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{2 x^{2} - 9 x + 4 + 3 \cdot 2 (x \cdot x)}{2 x (2 x - 1)}$

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 9 x + 4 + 3 \cdot 2 x^{2}}{2 x (2 x - 1)}$

Multiply $3$ by $2$ .

$\frac{2 x^{2} - 9 x + 4 + 6 x^{2}}{2 x (2 x - 1)}$

Add $2 x^{2}$ and $6 x^{2}$ .

$\frac{8 x^{2} - 9 x + 4}{2 x (2 x - 1)}$

Simplify (x-4)/(2x)+(3x)/(2x-1)

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