To solve the equation [tex]\(9|x-1| = -45\)[/tex], let's carefully analyze each step and the properties of the absolute value.
1. Understand the Absolute Value Function:
- The absolute value function [tex]\( |x-1| \)[/tex] represents the distance of [tex]\( x-1 \)[/tex] from 0 on the number line. It is always non-negative, meaning [tex]\( |x-1| \geq 0 \)[/tex] for all real numbers [tex]\( x \)[/tex].
2. Consider the Multiplication by 9:
- Multiplying the absolute value by 9 results in [tex]\( 9|x-1| \)[/tex]. Since [tex]\( |x-1| \geq 0 \)[/tex], this expression will also be non-negative, i.e., [tex]\( 9|x-1| \geq 0 \)[/tex].
3. Analyze the Given Equation:
- The equation is [tex]\( 9|x-1| = -45 \)[/tex]. The left side, [tex]\( 9|x-1| \)[/tex], as we have established, is non-negative.
- The right side of the equation is -45, which is a negative number.
4. Equating Non-negative and Negative Values:
- Since [tex]\( 9|x-1| \)[/tex] is non-negative and [tex]\(-45\)[/tex] is negative, it is impossible for a non-negative number to equal a negative number. This violates the basic properties of real numbers.
5. Conclusion:
- Because a non-negative value can never be equal to a negative value, the equation [tex]\( 9|x-1| = -45 \)[/tex] has no solutions. There are no values of [tex]\( x \)[/tex] that can satisfy this equation.
Therefore, the correct answer is:
[tex]\[
\boxed{\text{A. No Solution}}
\][/tex]
