To solve the inequality [tex]\( |x - 2| + 4 \leq 3 \)[/tex]:
1. Start by isolating the absolute value expression:
[tex]\[
|x - 2| + 4 \leq 3
\][/tex]
Subtract 4 from both sides:
[tex]\[
|x - 2| \leq 3 - 4
\][/tex]
Simplifying the right side:
[tex]\[
|x - 2| \leq -1
\][/tex]
2. Next, consider the properties of absolute values. The absolute value expression [tex]\( |x - 2| \)[/tex] represents the distance between [tex]\( x \)[/tex] and 2 on the number line, which is always non-negative (i.e., [tex]\( |x - 2| \geq 0 \)[/tex] for all [tex]\( x \)[/tex]).
3. The inequality [tex]\( |x - 2| \leq -1 \)[/tex] suggests that an absolute value must be less than or equal to a negative number. Since an absolute value cannot be negative, it is impossible for any [tex]\( x \)[/tex] to satisfy this condition:
[tex]\[
|x - 2| \leq -1
\][/tex]
Thus, there are no values of [tex]\( x \)[/tex] that can make this inequality true.
Therefore, the solution to the inequality [tex]\( |x - 2| + 4 \leq 3 \)[/tex] is:
[tex]\[
\boxed{\text{no solution}}
\][/tex]
