Question

Is the Equation an Identity? Determine whether the given equation is an identity. If the equation is not an identity, find all its solutions.

$\frac{\mathbf{x}}{\mathbf{x}\mathbf{+}\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{+}\mathit{x}$

Short answer

$\begin{array}{l}x=\frac{-1+\sqrt{3}i}{2}\\ x=\frac{-1-\sqrt{3i}}{2}\end{array}\mathbb{Z}$

Step-by-step solution

Step 1.Given information.

The given equation, $\frac{x}{x+1}=1+x$.

Step 2. Given equation is an identity or not and if it is not identity then find the solutions

We have to solve this equation by cross multiplying to get solution if is it is not an identity.

To check $\frac{x}{1+x}=1+x$is whether equation is identity or not

If it is an identity it is true for every number
we check it by putting 1 in equation is identity or not
Put $x=1$in the given equation.

$\frac{1}{1+1}\ne 1+1$

Hence, $LHS\ne \text{RHS}$so it not an identity so we have find the solution of it.

Multiplying $(1+x)$both sides, we get:-

$\begin{array}{l}x=(1+x)(1+x)\\ x=1+x+x+x^2\\ x=1+2x+x^2\\ x^2+x+1=0\end{array}$

Solution of the quadratic equation is:-

$\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-1\pm \sqrt{1^2-4\cdot 1\cdot 1}}{2\cdot 1}\\ x=\frac{-1\pm \sqrt{1-4}}{2}\\ x=\frac{-1\pm \sqrt{-3}}{2}\\ x=\frac{-1\pm \sqrt{3}i}{2}\end{array}$

Hence, solution are:-

$\begin{array}{l}x=\frac{-1+\sqrt{3}i}{2}\\ x=\frac{-1-\sqrt{3i}}{2}\end{array}$