Solve x-1/x-2-x+6/x+5=7/(x-2)(x+5)

$$\frac { x - 1 } { x - 2 } - \frac { x + 6 } { x + 5 } = \frac { 7 } { ( x - 2 ) ( x + 5 ) }$$

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus 2,-5$$

Solution Steps

Variable x cannot be equal to any of the values -5,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x-2,x+5,\left(x-2\right)\left(x+5\right).

$$\left(x+5\right)\left(x-1\right)-\left(x-2\right)\left(x+6\right)=7$$

Use the distributive property to multiply x+5 by x-1 and combine like terms.

$$x^{2}+4x-5-\left(x-2\right)\left(x+6\right)=7$$

Use the distributive property to multiply x-2 by x+6 and combine like terms.

$$x^{2}+4x-5-\left(x^{2}+4x-12\right)=7$$

To find the opposite of x^{2}+4x-12, find the opposite of each term.

$$x^{2}+4x-5-x^{2}-4x+12=7$$

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

$$4x-5-4x+12=7$$

Combine 4x and -4x to get 0.

$$-5+12=7$$

Add -5 and 12 to get 7.

$$7=7$$

Compare 7 and 7.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Variable x cannot be equal to any of the values -5,2.

$$x\in \mathrm{C}\setminus -5,2$$

Solve for x

$$x\in \mathrm{R}\setminus 2,-5$$

Solution Steps

Variable x cannot be equal to any of the values -5,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x-2,x+5,\left(x-2\right)\left(x+5\right).

$$\left(x+5\right)\left(x-1\right)-\left(x-2\right)\left(x+6\right)=7$$

Use the distributive property to multiply x+5 by x-1 and combine like terms.

$$x^{2}+4x-5-\left(x-2\right)\left(x+6\right)=7$$

Use the distributive property to multiply x-2 by x+6 and combine like terms.

$$x^{2}+4x-5-\left(x^{2}+4x-12\right)=7$$

To find the opposite of x^{2}+4x-12, find the opposite of each term.

$$x^{2}+4x-5-x^{2}-4x+12=7$$

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

$$4x-5-4x+12=7$$

Combine 4x and -4x to get 0.

$$-5+12=7$$

Add -5 and 12 to get 7.

$$7=7$$

Compare 7 and 7.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to any of the values -5,2.

$$x\in \mathrm{R}\setminus -5,2$$