To solve the equation [tex]\( |4w - 1| = 5w + 37 \)[/tex], we need to consider the definition and properties of absolute value. The absolute value function [tex]\( |x| \)[/tex] is defined as:
[tex]\[
|x| = \left\{
\begin{array}{ll}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{array}
\right.
\][/tex]
Thus, the equation [tex]\( |4w - 1| = 5w + 37 \)[/tex] splits into two cases:
### Case 1: [tex]\(4w - 1 \geq 0\)[/tex]
In this scenario, the absolute value function can be simplified as:
[tex]\[ |4w - 1| = 4w - 1 \][/tex]
Substituting this into the equation, we get:
[tex]\[ 4w - 1 = 5w + 37 \][/tex]
Now, solve for [tex]\( w \)[/tex]:
[tex]\[
4w - 1 = 5w + 37
\][/tex]
[tex]\[
4w - 5w = 37 + 1
\][/tex]
[tex]\[
-w = 38
\][/tex]
[tex]\[
w = -38
\][/tex]
Next, we need to check if this solution is valid given the constraint [tex]\( 4w - 1 \geq 0 \)[/tex]:
[tex]\[
4(-38) - 1 \geq 0
\][/tex]
[tex]\[
-152 - 1 \geq 0
\][/tex]
[tex]\[
-153 \geq 0 \quad \text{(False)}
\][/tex]
Since [tex]\(-38\)[/tex] does not satisfy the condition [tex]\( 4w - 1 \geq 0 \)[/tex], this solution is invalid.
### Case 2: [tex]\(4w - 1 < 0\)[/tex]
In this case, the absolute value function can be simplified as:
[tex]\[ |4w - 1| = -(4w - 1) \][/tex]
Substituting this into the equation, we get:
[tex]\[ -(4w - 1) = 5w + 37 \][/tex]
[tex]\[ -4w + 1 = 5w + 37 \][/tex]
Now, solve for [tex]\( w \)[/tex]:
[tex]\[
1 = 9w + 37
\][/tex]
[tex]\[
1 - 37 = 9w
\][/tex]
[tex]\[
-36 = 9w
\][/tex]
[tex]\[
w = -4
\][/tex]
Next, we need to check if this solution is valid given the constraint [tex]\( 4w - 1 < 0 \)[/tex]:
[tex]\[
4(-4) - 1 < 0
\][/tex]
[tex]\[
-16 - 1 < 0
\][/tex]
[tex]\[
-17 < 0 \quad \text{(True)}
\][/tex]
Since [tex]\(-4\)[/tex] satisfies the condition [tex]\( 4w - 1 < 0 \)[/tex], it is a valid solution.
### Conclusion
The solution to the equation [tex]\( |4w - 1| = 5w + 37 \)[/tex] is:
[tex]\[
w = -4
\][/tex]
