Since $\frac{x}{x}$ and $\frac{1}{x}$ have the same denominator, add them by adding their numerators.
$$1+\frac{1}{1+\frac{1}{\frac{x+1}{x}}}=5$$
Variable $x$ cannot be equal to $0$ since division by zero is not defined. Divide $1$ by $\frac{x+1}{x}$ by multiplying $1$ by the reciprocal of $\frac{x+1}{x}$.
$$1+\frac{1}{1+\frac{x}{x+1}}=5$$
To add or subtract expressions, expand them to make their denominators the same. Multiply $1$ times $\frac{x+1}{x+1}$.
$$1+\frac{1}{\frac{x+1}{x+1}+\frac{x}{x+1}}=5$$
Since $\frac{x+1}{x+1}$ and $\frac{x}{x+1}$ have the same denominator, add them by adding their numerators.
$$1+\frac{1}{\frac{x+1+x}{x+1}}=5$$
Combine like terms in $x+1+x$.
$$1+\frac{1}{\frac{2x+1}{x+1}}=5$$
Variable $x$ cannot be equal to $-1$ since division by zero is not defined. Divide $1$ by $\frac{2x+1}{x+1}$ by multiplying $1$ by the reciprocal of $\frac{2x+1}{x+1}$.
$$1+\frac{x+1}{2x+1}=5$$
To add or subtract expressions, expand them to make their denominators the same. Multiply $1$ times $\frac{2x+1}{2x+1}$.
$$\frac{2x+1}{2x+1}+\frac{x+1}{2x+1}=5$$
Since $\frac{2x+1}{2x+1}$ and $\frac{x+1}{2x+1}$ have the same denominator, add them by adding their numerators.
$$\frac{2x+1+x+1}{2x+1}=5$$
Combine like terms in $2x+1+x+1$.
$$\frac{3x+2}{2x+1}=5$$
Variable $x$ cannot be equal to $-\frac{1}{2}$ since division by zero is not defined. Multiply both sides of the equation by $2x+1$.
$$3x+2=5\left(2x+1\right)$$
Use the distributive property to multiply $5$ by $2x+1$.
$$3x+2=10x+5$$
Subtract $10x$ from both sides.
$$3x+2-10x=5$$
Combine $3x$ and $-10x$ to get $-7x$.
$$-7x+2=5$$
Subtract $2$ from both sides.
$$-7x=5-2$$
Subtract $2$ from $5$ to get $3$.
$$-7x=3$$
Divide both sides by $-7$.
$$x=\frac{3}{-7}$$
Fraction $\frac{3}{-7}$ can be rewritten as $-\frac{3}{7}$ by extracting the negative sign.
