To solve the equation [tex]\( |x + 6| = -8 \)[/tex], let's carefully analyze the properties of absolute value expressions.
1. Understanding Absolute Value:
The absolute value of a number, denoted by [tex]\( |a| \)[/tex], is defined as the distance of [tex]\( a \)[/tex] from zero on the number line. It always results in a non-negative number, meaning [tex]\( |a| \geq 0 \)[/tex] for any real number [tex]\( a \)[/tex].
2. Equation Analysis:
Given the equation [tex]\( |x + 6| = -8 \)[/tex]:
- We need to recognize that an absolute value expression [tex]\( |x + 6| \)[/tex] can never be negative.
- Therefore, the equation [tex]\( |x + 6| = -8 \)[/tex] is asking us to find a value of [tex]\( x \)[/tex] such that the non-negative absolute value of [tex]\( x + 6 \)[/tex] is equal to [tex]\(-8\)[/tex].
3. Checking for Solutions:
- Since the absolute value cannot be negative, the equation [tex]\( |x + 6| = -8 \)[/tex] does not make any sense in the set of real numbers.
- There are no real numbers [tex]\( x \)[/tex] such that the absolute value of [tex]\( x + 6 \)[/tex] equals a negative number.
4. Conclusion:
Given the properties of absolute values, it is clear that there is no solution to the equation [tex]\( |x + 6| = -8 \)[/tex].
Hence, the equation [tex]\( |x + 6| = -8 \)[/tex] has no solution.
