Q. 20 - Calculus

Question

Use algebra to solve the inequality $0<\left|x-c\right|<\delta$ and show that its solution set is $x\in \left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$.

Step-by-Step Solution

Answer

On solving the given inequality, we get,

$x\in \left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$

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,

$0<\left|x-c\right|<\delta$

Then, its solution set is $x\in \left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$.

Step 2. Solve the inequation when greater than zero.

Consider the inequality,

For $x-c>0$,

$$\begin{gathered} 0<\left|x-c\right|<\delta \\ 0<x-c<\delta \end{gathered}$$

Add c on every sides,

$$\begin{gathered} 0+c<c+x-c<\delta +c \\ c<x<\delta +c \end{gathered}$$

Step 3. Solve the inequation when less than zero.

Consider the inequality,

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

$$\begin{gathered} 0<\left|x-c\right|<\delta \\ 0<-\left(x-c\right)<\delta \\ -\delta <x-c<0 \end{gathered}$$

Add c on every sides,

$$\begin{gathered} -\delta +c<c+x-c<0+c \\ c-\delta <x<c \end{gathered}$$

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