To solve the equation [tex]\( |x + 4| = -6 \)[/tex], we need to understand the properties of absolute value functions. The absolute value of a number is always non-negative, meaning it can never be less than zero. It represents the distance of a number from zero on the number line, which is always a positive value or zero.
Given the equation [tex]\( |x + 4| = -6 \)[/tex]:
1. Absolute Value Properties: The absolute value, denoted by the vertical bars [tex]\( |\cdot| \)[/tex], is defined as:
- [tex]\( |a| = a \)[/tex] if [tex]\( a \geq 0 \)[/tex]
- [tex]\( |a| = -a \)[/tex] if [tex]\( a < 0 \)[/tex]
2. Assessment of the Given Equation: Here, the right side of the equation is [tex]\(-6\)[/tex]. Absolute value expressions can never be negative. Hence, an equation where [tex]\( |x + 4| \)[/tex] is set equal to a negative number has no valid solutions.
Since the left side of the equation involves the absolute value of [tex]\( x + 4 \)[/tex], and the absolute value cannot be equal to a negative number, there are no real numbers [tex]\( x \)[/tex] that satisfy this equation.
Thus, the solution to the equation [tex]\( |x + 4| = -6 \)[/tex] is:
A. No solution.
So, the answer is:
A. No solution.
