Question

Solve for $x$: $\dfrac{3}{{x - 1}} + \dfrac{1}{{x + 1}} = \dfrac{4}{x}$ , where $x \ne 0$,$x \ne 1$ , $x \ne - 1$ .

Answer 1

Hint: An algebraic expression is an expression where the variables and the constants are combined together. Simplification of an algebraic expression is the process of writing the given algebraic expression in an effective way and in the most comfortable way to understand without affecting the original expression. Moreover, there are various steps that are involved to simplify an algebraic expression.
Formula used:
$\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$

Complete step by step answer:
 The given algebraic equation is$\dfrac{3}{{x - 1}} + \dfrac{1}{{x + 1}} = \dfrac{4}{x}$
Taking LCM for the expressions on left side, we have
$\dfrac{{3\left( {x + 1} \right) + 1\left( {x - 1} \right)}}{{\left( {x + 1} \right)\left( {x - 1} \right)}} = \dfrac{4}{x}$
Now, we need to use the given formula on denominator, we get
$\dfrac{{3x + 3 + x - 1}}{{\left( {{x^2} - 1} \right)}} = \dfrac{4}{x}$
$\dfrac{{4x + 2}}{{\left( {{x^2} - 1} \right)}} = \dfrac{4}{x}$
Next, we need to cross-multiply the terms on both sides,
$\left( {4x + 2} \right)x = 4\left( {{x^2} - 1} \right)$
$4{x^2} + 2x = 4{x^2} - 4$
On further solving, wehave
$4{x^2} + 2x - 4{x^2} + 4 = 0$
$2x + 4 = 0$
$2x = - 4$
$x = - \dfrac{4}{2}$
$x = - 2$ it is the required solution.
$x = - 2$ is the solution for the given algebraic equation$\dfrac{3}{{x - 1}} + \dfrac{1}{{x + 1}} = \dfrac{4}{x}$

Note: Simplification of an expression is the process of changing the expression in an effective manner without changing the meaning of an expression. An algebraic expression is an expression where the variables and the constants are combined together
Moreover, there are various steps that are involved to simplify an algebraic expression. Some of the steps are listed below:
If the given algebraic expression contains like terms, we need to combine them.
Example: $3x + 2x + 4 = 5x + 4$
We need to split an algebraic expression into factors (i.e) the process of finding the factors for the given expression.
Example: ${x^2} + 4x + 3 = (x + 3)(x + 1)$
We need to expand an algebraic expression (i.e) we have to remove the respective brackets of an expression.
Example: $3(a + b) = 3a + 3b$.
We need to cancel out the common terms in an algebraic expression.
Example: $\dfrac{{{x^2} + 4x + 3}}{{x + 1}} = \dfrac{{(x + 3)(x + 1)}}{{x + 1}}$
$ = x + 3$