Q. 21 - Calculus

Question

Use algebra to solve the inequality $\left|f\left(x\right)-L\right|<\in$and show that its solution set is $f\left(x\right)\in \left(L-\in ,L+\in \right)$.

Step-by-Step Solution

Answer

On solving the given inequality, we get,

$f\left(x\right)\in \left(L-\in ,L+\in \right)$

To obtain the solution set, first we need to solve the inequation when greater than and less than zero.

Step 1. Given information.

Consider the given question,

$\left|f\left(x\right)-L\right|<\in$

Then, its solution set is $f\left(x\right)\in \left(L-\in ,L+\in \right)$.

Step 2. Solve the inequation when greater than zero.

Consider the inequality,

For $f\left(x\right)-L>0$,

$$\begin{gathered} \left|f\left(x\right)-L\right|<\in \\ f\left(x\right)-L<\in \end{gathered}$$

Add L on every side,

$$\begin{gathered} f\left(x\right)-L+L<\in +L \\ f\left(x\right)>\in \end{gathered}$$

Step 3. Solve the inequation when less than zero.

Consider the inequality,

For $f\left(x\right)-L<0$,

$$\begin{gathered} \left|f\left(x\right)-L\right|<\in \\ -\left[f\left(x\right)-L\right]<\in \\ -f\left(x\right)+L<\in \\ f\left(x\right)>L-\in \end{gathered}$$

Hence, combining both the results the solution can be written as $f\left(x\right)\in \left(L-\in ,L+\in \right)$.