Solve by Factoring 1/(x-2)+3/(x+3)=4/(x^2+x-6)

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

Step 1

Subtract $\frac{4}{x^{2} + x - 6}$ from both sides of the equation.

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

Step 2

Simplify $\frac{1}{x - 2} + \frac{3}{x + 3} - \frac{4}{x^{2} + x - 6}$ .

Factor $x^{2} + x - 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Write the factored form using these integers.

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Multiply $3$ by $- 2$ .

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

Add $x$ and $3 x$ .

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

Subtract $6$ from $3$ .

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

Subtract $4$ from $- 3$ .

$\frac{4 x - 7}{(x + 3) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$4 x - 7 = 0$

Step 4

Solve the equation for $x$ .

Add $7$ to both sides of the equation.

$4 x = 7$

Divide each term in $4 x = 7$ by $4$ and simplify.

Divide each term in $4 x = 7$ by $4$ .

$\frac{4 x}{4} = \frac{7}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{7}{4}$

Divide $x$ by $1$ .

$x = \frac{7}{4}$

Solve by Factoring 1/(x-2)+3/(x+3)=4/(x^2+x-6)

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