Question

The solution set of inequality \( |x+2|<3 \) is

• A. \( -1<x<5 \)
• B. \( -5<x<1 \)
• C. \( -1<x<3 \)
• D. \( 1<x<3 \)
• E. \( -1<x<1 \)

Hint:

For inequalities of the form \( |A|<k \), always convert them into the double inequality \( -k<A<k \). This is the fastest and safest method.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves solving a simple absolute value inequality. The inequality \(|u|<a\) (where \(a>0\)) means that the distance of u from 0 is less than a. This can be written as \(-a<u<a\).
Step 2: Key Formula or Approach:
Using the property \(|u|<a \iff -a<u<a\), we set \(u = x+2\) and \(a=3\).
The inequality \(|x+2|<3\) is equivalent to:
\[ -3<x+2<3 \] Step 3: Detailed Explanation:
We need to isolate x in the compound inequality \(-3<x+2<3\). To do this, we subtract 2 from all three parts of the inequality.
\[ -3 - 2<x + 2 - 2<3 - 2 \] \[ -5<x<1 \] This means x lies in the open interval \((-5, 1)\).
Step 4: Final Answer:
The solution set for the inequality is \(-5<x<1\). This corresponds to option (B).