To solve the inequality [tex]\( |x + 2| + 4 < 3 \)[/tex], let's go through it step by step.
1. Understanding the inequality:
We are given [tex]\( |x + 2| + 4 < 3 \)[/tex]. This is an absolute value inequality, which generally requires us to consider different cases based on the definition of absolute value. However, before we dive into cases, let's simplify the inequality.
2. Isolate the absolute value term:
To make the problem simpler, we want to isolate the absolute value term on one side of the inequality.
[tex]\[
|x + 2| + 4 < 3
\][/tex]
Subtract 4 from both sides:
[tex]\[
|x + 2| + 4 - 4 < 3 - 4
\][/tex]
[tex]\[
|x + 2| < -1
\][/tex]
3. Analyzing the inequality:
Now, we need to consider what [tex]\( |x + 2| < -1 \)[/tex] means. The absolute value function [tex]\( |y| \)[/tex] for any real number [tex]\( y \)[/tex] is always non-negative. That means [tex]\( |x + 2| \geq 0 \)[/tex] for all [tex]\( x \)[/tex].
4. Checking for solutions:
Since [tex]\( |x + 2| \)[/tex] is always non-negative, [tex]\( |x + 2| < -1 \)[/tex] implies that we need a value less than -1. It's clear that no real number [tex]\( x \)[/tex] will satisfy this inequality because an absolute value cannot be negative, let alone less than a negative number.
5. Conclusion:
Given that [tex]\( |x + 2| \)[/tex] cannot be less than -1, there are no real numbers [tex]\( x \)[/tex] that satisfy the inequality [tex]\( |x + 2| + 4 < 3 \)[/tex].
Therefore, the solution to the inequality [tex]\( |x + 2| + 4 < 3 \)[/tex] is the empty set. In other words, there are no values of [tex]\( x \)[/tex] that make this inequality true.
