Solve 1/(x + 1) - 1/(x + 2) = 1/(x + 3) - 1/(x + 4) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Solve for x

$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$

Steps for Solving Linear Equation

Variable $x$ cannot be equal to any of the values $-4,-3,-2,-1$ since division by zero is not defined. Multiply both sides of the equation by $\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)$, the least common multiple of $x+1,x+2,x+3,x+4$.

$$\left(x+2\right)\left(x+3\right)\left(x+4\right)-\left(x+1\right)\left(x+3\right)\left(x+4\right)=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

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

$$\left(x^{2}+5x+6\right)\left(x+4\right)-\left(x+1\right)\left(x+3\right)\left(x+4\right)=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $x^{2}+5x+6$ by $x+4$ and combine like terms.

$$x^{3}+9x^{2}+26x+24-\left(x+1\right)\left(x+3\right)\left(x+4\right)=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $x+1$ by $x+3$ and combine like terms.

$$x^{3}+9x^{2}+26x+24-\left(x^{2}+4x+3\right)\left(x+4\right)=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $x^{2}+4x+3$ by $x+4$ and combine like terms.

$$x^{3}+9x^{2}+26x+24-\left(x^{3}+8x^{2}+19x+12\right)=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

To find the opposite of $x^{3}+8x^{2}+19x+12$, find the opposite of each term.

$$x^{3}+9x^{2}+26x+24-x^{3}-8x^{2}-19x-12=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Combine $x^{3}$ and $-x^{3}$ to get $0$.

$$9x^{2}+26x+24-8x^{2}-19x-12=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Combine $9x^{2}$ and $-8x^{2}$ to get $x^{2}$.

$$x^{2}+26x+24-19x-12=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Combine $26x$ and $-19x$ to get $7x$.

$$x^{2}+7x+24-12=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Subtract $12$ from $24$ to get $12$.

$$x^{2}+7x+12=\left(x+1\right)\left(x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $x+1$ by $x+2$ and combine like terms.

$$x^{2}+7x+12=\left(x^{2}+3x+2\right)\left(x+4\right)-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $x^{2}+3x+2$ by $x+4$ and combine like terms.

$$x^{2}+7x+12=x^{3}+7x^{2}+14x+8-\left(x+1\right)\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $x+1$ by $x+2$ and combine like terms.

$$x^{2}+7x+12=x^{3}+7x^{2}+14x+8-\left(x^{2}+3x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $x^{2}+3x+2$ by $x+3$ and combine like terms.

$$x^{2}+7x+12=x^{3}+7x^{2}+14x+8-\left(x^{3}+6x^{2}+11x+6\right)$$

To find the opposite of $x^{3}+6x^{2}+11x+6$, find the opposite of each term.

$$x^{2}+7x+12=x^{3}+7x^{2}+14x+8-x^{3}-6x^{2}-11x-6$$

Combine $x^{3}$ and $-x^{3}$ to get $0$.

$$x^{2}+7x+12=7x^{2}+14x+8-6x^{2}-11x-6$$

Combine $7x^{2}$ and $-6x^{2}$ to get $x^{2}$.

$$x^{2}+7x+12=x^{2}+14x+8-11x-6$$

Combine $14x$ and $-11x$ to get $3x$.

$$x^{2}+7x+12=x^{2}+3x+8-6$$

Subtract $6$ from $8$ to get $2$.

$$x^{2}+7x+12=x^{2}+3x+2$$

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

$$x^{2}+7x+12-x^{2}=3x+2$$

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

$$7x+12=3x+2$$

Subtract $3x$ from both sides.

$$7x+12-3x=2$$

Combine $7x$ and $-3x$ to get $4x$.

$$4x+12=2$$

Subtract $12$ from both sides.

$$4x=2-12$$

Subtract $12$ from $2$ to get $-10$.

$$4x=-10$$

Divide both sides by $4$.

$$x=\frac{-10}{4}$$

Reduce the fraction $\frac{-10}{4}$ to lowest terms by extracting and canceling out $2$.

$$x=-\frac{5}{2}$$