Solve operatorname{ion}1/x+4-1/x-7=11/30

$$\operatorname { ion } \frac { 1 } { x + 4 } - \frac { 1 } { x - 7 } = \frac { 11 } { 30 }$$

Solve for n

$$\left\{\begin{matrix}n=\frac{\left(11x-47\right)\left(x+4\right)}{30o\left(ix-7i\right)}\text{, }&x\neq 7\text{ and }o\neq 0\text{ and }x\neq -4\\n\in \mathrm{C}\text{, }&x=\frac{47}{11}\text{ and }o=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30\left(x-7\right)\left(x+4\right), the least common multiple of x+4,x-7,30.

$$ion\left(30x-210\right)\times 1-\left(30x+120\right)=11\left(x-7\right)\left(x+4\right)$$

Multiply i and 1 to get i.

$$ion\left(30x-210\right)-\left(30x+120\right)=11\left(x-7\right)\left(x+4\right)$$

Use the distributive property to multiply ion by 30x-210.

$$30inox-210ino-\left(30x+120\right)=11\left(x-7\right)\left(x+4\right)$$

To find the opposite of 30x+120, find the opposite of each term.

$$30inox-210ino-30x-120=11\left(x-7\right)\left(x+4\right)$$

Use the distributive property to multiply 11 by x-7.

$$30inox-210ino-30x-120=\left(11x-77\right)\left(x+4\right)$$

Use the distributive property to multiply 11x-77 by x+4 and combine like terms.

$$30inox-210ino-30x-120=11x^{2}-33x-308$$

Add 30x to both sides.

$$30inox-210ino-120=11x^{2}-33x-308+30x$$

Combine -33x and 30x to get -3x.

$$30inox-210ino-120=11x^{2}-3x-308$$

Add 120 to both sides.

$$30inox-210ino=11x^{2}-3x-308+120$$

Add -308 and 120 to get -188.

$$30inox-210ino=11x^{2}-3x-188$$

Combine all terms containing n.

$$\left(30iox-210io\right)n=11x^{2}-3x-188$$

Divide both sides by -210io+30iox.

$$\frac{\left(30iox-210io\right)n}{30iox-210io}=\frac{\left(11x-47\right)\left(x+4\right)}{30iox-210io}$$

Dividing by -210io+30iox undoes the multiplication by -210io+30iox.

$$n=\frac{\left(11x-47\right)\left(x+4\right)}{30iox-210io}$$

Divide \left(-47+11x\right)\left(4+x\right) by -210io+30iox.

$$n=\frac{\left(11x-47\right)\left(x+4\right)}{30o\left(ix-7i\right)}$$

Solve for o

$$\left\{\begin{matrix}o=\frac{\left(11x-47\right)\left(x+4\right)}{30n\left(ix-7i\right)}\text{, }&x\neq 7\text{ and }n\neq 0\text{ and }x\neq -4\\o\in \mathrm{C}\text{, }&x=\frac{47}{11}\text{ and }n=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30\left(x-7\right)\left(x+4\right), the least common multiple of x+4,x-7,30.

$$ion\left(30x-210\right)\times 1-\left(30x+120\right)=11\left(x-7\right)\left(x+4\right)$$

Multiply i and 1 to get i.

$$ion\left(30x-210\right)-\left(30x+120\right)=11\left(x-7\right)\left(x+4\right)$$

Use the distributive property to multiply ion by 30x-210.

$$30inox-210ino-\left(30x+120\right)=11\left(x-7\right)\left(x+4\right)$$

To find the opposite of 30x+120, find the opposite of each term.

$$30inox-210ino-30x-120=11\left(x-7\right)\left(x+4\right)$$

Use the distributive property to multiply 11 by x-7.

$$30inox-210ino-30x-120=\left(11x-77\right)\left(x+4\right)$$

Use the distributive property to multiply 11x-77 by x+4 and combine like terms.

$$30inox-210ino-30x-120=11x^{2}-33x-308$$

Add 30x to both sides.

$$30inox-210ino-120=11x^{2}-33x-308+30x$$

Combine -33x and 30x to get -3x.

$$30inox-210ino-120=11x^{2}-3x-308$$

Add 120 to both sides.

$$30inox-210ino=11x^{2}-3x-308+120$$

Add -308 and 120 to get -188.

$$30inox-210ino=11x^{2}-3x-188$$

Combine all terms containing o.

$$\left(30inx-210in\right)o=11x^{2}-3x-188$$

Divide both sides by -210in+30inx.

$$\frac{\left(30inx-210in\right)o}{30inx-210in}=\frac{\left(11x-47\right)\left(x+4\right)}{30inx-210in}$$

Dividing by -210in+30inx undoes the multiplication by -210in+30inx.

$$o=\frac{\left(11x-47\right)\left(x+4\right)}{30inx-210in}$$

Divide \left(-47+11x\right)\left(4+x\right) by -210in+30inx.

$$o=\frac{\left(11x-47\right)\left(x+4\right)}{30n\left(ix-7i\right)}$$