To solve the equation [tex]\(3|4w - 1| - 5 = 10\)[/tex], follow these detailed steps:
1. Isolate the absolute value term:
Start by isolating [tex]\( |4w - 1| \)[/tex]. Add 5 to both sides to accomplish this:
[tex]\[
3|4w - 1| - 5 + 5 = 10 + 5
\][/tex]
Simplifying this yields:
[tex]\[
3|4w - 1| = 15
\][/tex]
2. Solve for the absolute value expression:
Next, divide both sides by 3 to solve for [tex]\( |4w - 1| \)[/tex]:
[tex]\[
\frac{3|4w - 1|}{3} = \frac{15}{3}
\][/tex]
This simplifies to:
[tex]\[
|4w - 1| = 5
\][/tex]
3. Break down the absolute value equation:
The absolute value equation [tex]\( |4w - 1| = 5 \)[/tex] can be split into two separate linear equations:
[tex]\[
4w - 1 = 5 \quad \text{and} \quad 4w - 1 = -5
\][/tex]
4. Solve each of the linear equations:
- For the first equation [tex]\(4w - 1 = 5\)[/tex]:
[tex]\[
4w - 1 = 5
\][/tex]
Add 1 to both sides:
[tex]\[
4w = 6
\][/tex]
Divide by 4:
[tex]\[
w = \frac{6}{4} = 1.5
\][/tex]
- For the second equation [tex]\(4w - 1 = -5\)[/tex]:
[tex]\[
4w - 1 = -5
\][/tex]
Add 1 to both sides:
[tex]\[
4w = -4
\][/tex]
Divide by 4:
[tex]\[
w = \frac{-4}{4} = -1.0
\][/tex]
5. Combine the solutions:
The two solutions to the original equation are:
[tex]\[
w = 1.5 \quad \text{and} \quad w = -1.0
\][/tex]
Therefore, the solutions to the equation [tex]\(3|4w - 1| - 5 = 10\)[/tex] are:
[tex]\[ w = 1.5 \quad \text{and} \quad w = -1.0 \][/tex]
