Question

Determine whether the given value is a solution of the equation

$\frac{1}{x}-\frac{1}{x-4}=1$

(a)$x=2$

(b)$x=4$

Short answer

(a) $x=2$ isa solution

(b) $x=4$ is not a solution

Step-by-step solution

Part a. Step 1. Apply the concept of solutions.

A value is the solution of given equation if upon substituting the value, the equation is satisfied.

Part a. Step 2. Analyze the given expression.

Given

$\frac{1}{x}-\frac{1}{x-4}=1$

To check whether the given value is solution of the equation, we substitute it in place of x.

Part a. Step 3. Solve.

$\frac{1}{x}-\frac{1}{x-4}=1$

Substitute $x=2$

Left Hand Side:

$\begin{array}{l}\frac{1}{x}-\frac{1}{x-4}\\ =\frac{1}{2}-\frac{1}{2-4}\\ =\frac{1}{2}+\frac{1}{2}\\ =1\end{array}$

Right Hand Side:

$1$

Since Left Hand Side is equal to Right Hand Side, $x=2$ is a solution of the equation.

Part b. Step 1. Apply the concept of solutions.

A value is the solution of given equation if upon substituting the value, the equation is satisfied.

Part b. Step 2. Analyze the given expression.

Given

$1-\left[2-\left(3-x\right)\right]=4x-\left(6+x\right)$

To check whether the given value is solution of the equation, we substitute it in place of x.

Part b. Step 3. Solve.

$\frac{1}{x}-\frac{1}{x-4}=1$

Substitute $x=4$

Left Hand Side:

$\begin{array}{l}\frac{1}{x}-\frac{1}{x-4}\\ =\frac{1}{4}-\frac{1}{4-4}\\ =\frac{1}{4}-\frac{1}{0}\\ Undefined\end{array}$

So, $x=4$ is not a solution of the equation.