To solve the inequality [tex]\(|x - 2| + 4 \leq 3\)[/tex], follow these steps:
1. Isolate the Absolute Value:
Begin by isolating the absolute value expression on one side of the inequality.
[tex]\[
|x - 2| + 4 \leq 3
\][/tex]
Subtract 4 from both sides of the inequality to isolate the absolute value:
[tex]\[
|x - 2| \leq 3 - 4
\][/tex]
This simplifies to:
[tex]\[
|x - 2| \leq -1
\][/tex]
2. Analyze the Absolute Value Inequality:
Recall that the absolute value of any real number is always non-negative (i.e., [tex]\(|x| \geq 0\)[/tex] for all [tex]\(x \in \mathbb{R}\)[/tex]). This means that [tex]\(|x - 2|\)[/tex] represents the distance between [tex]\(x\)[/tex] and 2 on the number line, which can never be negative. Therefore, there are no real numbers [tex]\(x\)[/tex] that satisfy an inequality where an absolute value is less than or equal to a negative number.
In our case:
[tex]\[
|x - 2| \leq -1
\][/tex]
Since [tex]\(|x - 2|\)[/tex] cannot be less than or equal to [tex]\(-1\)[/tex], there are no solutions to this inequality.
3. Conclusion:
Based on this analysis, the inequality [tex]\(|x - 2| + 4 \leq 3\)[/tex] has no solution. Therefore, the correct answer is:
[tex]\[
\boxed{\text{no solution}}
\][/tex]
