1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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Question

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

Solve for x

$x=5$

Steps for Solving Linear Equation

Variable $x$ cannot be equal to any of the values $2,3,4$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-4\right)\left(x-3\right)\left(x-2\right)$, the least common multiple of $x-3,x-2,x-4$.

$$\left(x-4\right)\left(x-2\right)\times 4-\left(x-4\right)\left(x-3\right)\times 3=\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply $x-4$ by $x-2$ and combine like terms.

$$\left(x^{2}-6x+8\right)\times 4-\left(x-4\right)\left(x-3\right)\times 3=\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply $x^{2}-6x+8$ by $4$.

$$4x^{2}-24x+32-\left(x-4\right)\left(x-3\right)\times 3=\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply $x-4$ by $x-3$ and combine like terms.

$$4x^{2}-24x+32-\left(x^{2}-7x+12\right)\times 3=\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply $x^{2}-7x+12$ by $3$.

$$4x^{2}-24x+32-\left(3x^{2}-21x+36\right)=\left(x-3\right)\left(x-2\right)$$

To find the opposite of $3x^{2}-21x+36$, find the opposite of each term.

$$4x^{2}-24x+32-3x^{2}+21x-36=\left(x-3\right)\left(x-2\right)$$

Combine $4x^{2}$ and $-3x^{2}$ to get $x^{2}$.

$$x^{2}-24x+32+21x-36=\left(x-3\right)\left(x-2\right)$$

Combine $-24x$ and $21x$ to get $-3x$.

$$x^{2}-3x+32-36=\left(x-3\right)\left(x-2\right)$$

Subtract $36$ from $32$ to get $-4$.

$$x^{2}-3x-4=\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply $x-3$ by $x-2$ and combine like terms.

$$x^{2}-3x-4=x^{2}-5x+6$$

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

$$x^{2}-3x-4-x^{2}=-5x+6$$

Combine $x^{2}$ and $-x^{2}$ to get $0$.

$$-3x-4=-5x+6$$

Add $5x$ to both sides.

$$-3x-4+5x=6$$

Combine $-3x$ and $5x$ to get $2x$.

$$2x-4=6$$

Add $4$ to both sides.

$$2x=6+4$$

Add $6$ and $4$ to get $10$.

$$2x=10$$

Divide both sides by $2$.

$$x=\frac{10}{2}$$

Divide $10$ by $2$ to get $5$.

$$x=5$$

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