Question

Solve the given inequality : \[1 \le \left| {x - 2} \right| \le 3\]

Answer 1

Hint: Here, we have to solve the inequation to find the solution set for the inequation. We will solve the inequalities taking into consideration two cases, i.e. if the variable is negative and if the variable is positive. Then we will combine the inequalities to find the solution set for the variable and hence the required answer.

Complete Step by Step Solution:
We are given with an inequality \[1 \le \left| {x - 2} \right| \le 3\].
First we will consider the case in the inequality that \[x - 2 < 0\].
\[x - 2 < 0\]
Adding 2 on both the sides, we get
\[ \Rightarrow x < 2\]
So, we can say that \[x\] might be negative.
Now, we will consider the inequality \[1 \le \left| {x - 2} \right| \le 3\].
Now, considering the variable to be negative, we get
\[1 \le - x + 2 \le 3\]
By subtracting \[2\] from the inequality, we get
\[ \Rightarrow 1 - 2 \le - x + 2 - 2 \le 3 - 2\]
\[ \Rightarrow - 1 \le - x \le 1\]
Since the variable is negative, we get
\[ \Rightarrow - 1 \ge - x \ge 1\]
Thus, \[x \in \left[ { - 1,1} \right]\] ……………………………………………………………………………………………………………\[\left( 1 \right)\]
Now, we will consider the case in the inequality that \[x - 2 > 0\] .
Adding 2 on both the sides, we get
\[ \Rightarrow x > 2\]
So, we can say that \[x\] will be positive.
Now, we will consider the inequality \[1 \le \left| {x - 2} \right| \le 3\].
Considering the variable to be positive, we get
\[ \Rightarrow 1 \le x - 2 \le 3\]
By adding \[2\] to the inequality, we get
\[ \Rightarrow 1 + 2 \le x - 2 + 2 \le 3 + 2\]
\[ \Rightarrow 3 \le x \le 5\]
Thus, \[x \in \left[ {3,5} \right]\] …………………………………………………………………………………………………………………..\[\left( 2 \right)\]
Combining equations \[\left( 1 \right)\] and \[\left( 2 \right)\] , we get
\[ \Rightarrow x \in \left[ { - 1,1} \right] \cup \left[ {3,5} \right]\]

Therefore, the solution set is \[x \in \left[ { - 1,1} \right] \cup \left[ {3,5} \right]\].

Note: We know that inequality is defined as a non-equal comparison between two numbers and two mathematical expressions with the variables which may be greater than or lesser than or greater than equal to or less than equal to. We should remember that adding the same quantity on both sides of an inequality does not change its direction. When we multiply or divide the positive number there is no change in the direction. When we multiply or divide by a negative number, there is a change in the direction of the inequality.