To solve the equation [tex]\( |x| + 19 = 12 \)[/tex], we'll proceed step-by-step:
1. Isolate the absolute value expression:
[tex]\[
|x| + 19 = 12
\][/tex]
Subtract 19 from both sides of the equation to isolate the absolute value.
[tex]\[
|x| = 12 - 19
\][/tex]
2. Simplify the right-hand side:
[tex]\[
|x| = -7
\][/tex]
3. Interpretation of the absolute value expression:
The absolute value of any real number can only be zero or a positive value. By definition, the absolute value function, [tex]\( |x| \)[/tex], represents the distance of [tex]\( x \)[/tex] from 0 on the number line, which cannot be negative.
Therefore, having [tex]\( |x| = -7 \)[/tex] is not possible because [tex]\(-7\)[/tex] is a negative number, and the absolute value function cannot yield a negative result.
4. Conclusion:
Since [tex]\( |x| = -7 \)[/tex] has no solutions in the real numbers, it follows that the original equation [tex]\( |x| + 19 = 12 \)[/tex] has no real solutions.
Thus, the conclusion is:
[tex]\[
\boxed{\text{No real solution}}
\][/tex]
