To solve the equation [tex]\( |x| + 2x = 3 \)[/tex], we need to consider the definition of the absolute value function and how it changes the equation based on the value of [tex]\( x \)[/tex]. Specifically, the absolute value function can be handled in a piecewise manner:
1. Case 1: [tex]\( x \geq 0 \)[/tex]
- For [tex]\( x \geq 0 \)[/tex], the absolute value function [tex]\( |x| \)[/tex] is simply [tex]\( x \)[/tex].
- Substituting [tex]\( |x| \)[/tex] with [tex]\( x \)[/tex] in the equation [tex]\( |x| + 2x = 3 \)[/tex], we get:
[tex]\[
x + 2x = 3
\][/tex]
- Simplifying the left-hand side, we have:
[tex]\[
3x = 3
\][/tex]
- Solving for [tex]\( x \)[/tex], we divide both sides by 3:
[tex]\[
x = 1
\][/tex]
- Since [tex]\( x = 1 \)[/tex] is indeed greater than or equal to 0, it is a valid solution for this case.
2. Case 2: [tex]\( x < 0 \)[/tex]
- For [tex]\( x < 0 \)[/tex], the absolute value function [tex]\( |x| \)[/tex] is [tex]\( -x \)[/tex].
- Substituting [tex]\( |x| \)[/tex] with [tex]\( -x \)[/tex] in the equation [tex]\( |x| + 2x = 3 \)[/tex], we get:
[tex]\[
-x + 2x = 3
\][/tex]
- Simplifying the left-hand side, we have:
[tex]\[
x = 3
\][/tex]
- Since [tex]\( x = 3 \)[/tex] is not less than 0, this solution is not valid for this case.
Summarizing both cases, we find that the only valid solution that satisfies the equation [tex]\( |x| + 2x = 3 \)[/tex] is:
[tex]\[
x = 1
\][/tex]
So the solution to the equation is:
[tex]\[
x = 1
\][/tex]
