Simplify (2x+3)/(x-4)-(x-5)/(x+2)

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

$\frac{(2 x + 3) (x + 2)}{(x - 4) (x + 2)} - \frac{(x - 5) (x - 4)}{(x + 2) (x - 4)}$

Reorder the factors of $(x - 4) (x + 2)$ .

$\frac{(2 x + 3) (x + 2)}{(x + 2) (x - 4)} - \frac{(x - 5) (x - 4)}{(x + 2) (x - 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(2 x + 3) (x + 2) - (x - 5) (x - 4)}{(x + 2) (x - 4)}$

Step 5

Simplify the numerator.

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

Apply the distributive property.

$\frac{2 x (x + 2) + 3 (x + 2) - (x - 5) (x - 4)}{(x + 2) (x - 4)}$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot 2 + 3 x + 3 \cdot 2 - (x - 5) (x - 4)}{(x + 2) (x - 4)}$

Multiply $2$ by $2$ .

$\frac{2 x^{2} + 4 x + 3 x + 3 \cdot 2 - (x - 5) (x - 4)}{(x + 2) (x - 4)}$

Multiply $3$ by $2$ .

$\frac{2 x^{2} + 4 x + 3 x + 6 - (x - 5) (x - 4)}{(x + 2) (x - 4)}$

Add $4 x$ and $3 x$ .

$\frac{2 x^{2} + 7 x + 6 - (x - 5) (x - 4)}{(x + 2) (x - 4)}$

Apply the distributive property.

$\frac{2 x^{2} + 7 x + 6 + (- x - - 5) (x - 4)}{(x + 2) (x - 4)}$

Multiply $- 1$ by $- 5$ .

$\frac{2 x^{2} + 7 x + 6 + (- x + 5) (x - 4)}{(x + 2) (x - 4)}$

Expand $(- x + 5) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 x^{2} + 7 x + 6 - x (x - 4) + 5 (x - 4)}{(x + 2) (x - 4)}$

Apply the distributive property.

$\frac{2 x^{2} + 7 x + 6 - x \cdot x - x \cdot - 4 + 5 (x - 4)}{(x + 2) (x - 4)}$

Apply the distributive property.

$\frac{2 x^{2} + 7 x + 6 - x \cdot x - x \cdot - 4 + 5 x + 5 \cdot - 4}{(x + 2) (x - 4)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{2 x^{2} + 7 x + 6 - (x \cdot x) - x \cdot - 4 + 5 x + 5 \cdot - 4}{(x + 2) (x - 4)}$

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 7 x + 6 - x^{2} - x \cdot - 4 + 5 x + 5 \cdot - 4}{(x + 2) (x - 4)}$

Multiply $- 4$ by $- 1$ .

$\frac{2 x^{2} + 7 x + 6 - x^{2} + 4 x + 5 x + 5 \cdot - 4}{(x + 2) (x - 4)}$

Multiply $5$ by $- 4$ .

$\frac{2 x^{2} + 7 x + 6 - x^{2} + 4 x + 5 x - 20}{(x + 2) (x - 4)}$

Add $4 x$ and $5 x$ .

$\frac{2 x^{2} + 7 x + 6 - x^{2} + 9 x - 20}{(x + 2) (x - 4)}$

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

$\frac{x^{2} + 7 x + 6 + 9 x - 20}{(x + 2) (x - 4)}$

Add $7 x$ and $9 x$ .

$\frac{x^{2} + 16 x + 6 - 20}{(x + 2) (x - 4)}$

Subtract $20$ from $6$ .

$\frac{x^{2} + 16 x - 14}{(x + 2) (x - 4)}$

Simplify (2x+3)/(x-4)-(x-5)/(x+2)

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