Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(2 x + 3) (2 x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(2 x - 3) (2 x + 3)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Rewrite using the commutative property of multiplication.

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

Step 5.3

Move $- 3$ to the left of $x$ .

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

Step 5.4

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

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Step 5.4.1

Move $x$ .

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

Step 5.4.2

Multiply $x$ by $x$ .

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

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

Step 5.5

Apply the distributive property.

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

Step 5.6

Multiply $2$ by $- 1$ .

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

Step 5.7

Multiply $- 1$ by $1$ .

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

Step 5.8

Expand $(- 2 x - 1) (2 x + 3)$ using the FOIL Method.

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Step 5.8.1

Apply the distributive property.

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

Step 5.8.2

Apply the distributive property.

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

Step 5.8.3

Apply the distributive property.

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

Step 5.9

Simplify and combine like terms.

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Step 5.9.1

Simplify each term.

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Step 5.9.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.9.1.2

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

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Step 5.9.1.2.1

Move $x$ .

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

Step 5.9.1.2.2

Multiply $x$ by $x$ .

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

Step 5.9.1.3

Multiply $- 2$ by $2$ .

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

Step 5.9.1.4

Multiply $3$ by $- 2$ .

$\frac{2 x^{2} - 3 x - 4 x^{2} - 6 x - 1 (2 x) - 1 \cdot 3}{(2 x + 3) (2 x - 3)}$

Step 5.9.1.5

Multiply $2$ by $- 1$ .

$\frac{2 x^{2} - 3 x - 4 x^{2} - 6 x - 2 x - 1 \cdot 3}{(2 x + 3) (2 x - 3)}$

Step 5.9.1.6

Multiply $- 1$ by $3$ .

$\frac{2 x^{2} - 3 x - 4 x^{2} - 6 x - 2 x - 3}{(2 x + 3) (2 x - 3)}$

Step 5.9.2

Subtract $2 x$ from $- 6 x$ .

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

Step 5.10

Subtract $4 x^{2}$ from $2 x^{2}$ .

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

Step 5.11

Subtract $8 x$ from $- 3 x$ .

$\frac{- 2 x^{2} - 11 x - 3}{(2 x + 3) (2 x - 3)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 2 x^{2}$ .

$\frac{- (2 x^{2}) - 11 x - 3}{(2 x + 3) (2 x - 3)}$

Step 6.2

Factor $- 1$ out of $- 11 x$ .

$\frac{- (2 x^{2}) - (11 x) - 3}{(2 x + 3) (2 x - 3)}$

Step 6.3

Factor $- 1$ out of $- (2 x^{2}) - (11 x)$ .

$\frac{- (2 x^{2} + 11 x) - 3}{(2 x + 3) (2 x - 3)}$

Step 6.4

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- (2 x^{2} + 11 x) - 1 (3)}{(2 x + 3) (2 x - 3)}$

Step 6.5

Factor $- 1$ out of $- (2 x^{2} + 11 x) - 1 (3)$ .

$\frac{- (2 x^{2} + 11 x + 3)}{(2 x + 3) (2 x - 3)}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (2 x^{2} + 11 x + 3)$ as $- 1 (2 x^{2} + 11 x + 3)$ .

$\frac{- 1 (2 x^{2} + 11 x + 3)}{(2 x + 3) (2 x - 3)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{2 x^{2} + 11 x + 3}{(2 x + 3) (2 x - 3)}$