Certainly! Let's solve for [tex]\( u \)[/tex] in the equation [tex]\( |2u - 5| = 7 \)[/tex].
### Step-by-Step Solution:
1. Understand the Absolute Value Equation:
The equation [tex]\( |2u - 5| = 7 \)[/tex] means that the expression inside the absolute value, [tex]\( 2u - 5 \)[/tex], can be either 7 or -7. This is because the absolute value of a number is that number without regard to its sign.
2. Set Up the Two Cases:
Case 1: [tex]\( 2u - 5 = 7 \)[/tex]
Case 2: [tex]\( 2u - 5 = -7 \)[/tex]
3. Solve Case 1 [tex]\( (2u - 5 = 7) \)[/tex]:
[tex]\[
2u - 5 = 7
\][/tex]
Add 5 to both sides to isolate the term with [tex]\( u \)[/tex]:
[tex]\[
2u - 5 + 5 = 7 + 5
\][/tex]
[tex]\[
2u = 12
\][/tex]
Divide both sides by 2:
[tex]\[
u = \frac{12}{2}
\][/tex]
[tex]\[
u = 6
\][/tex]
4. Solve Case 2 [tex]\( (2u - 5 = -7) \)[/tex]:
[tex]\[
2u - 5 = -7
\][/tex]
Add 5 to both sides to isolate the term with [tex]\( u \)[/tex]:
[tex]\[
2u - 5 + 5 = -7 + 5
\][/tex]
[tex]\[
2u = -2
\][/tex]
Divide both sides by 2:
[tex]\[
u = \frac{-2}{2}
\][/tex]
[tex]\[
u = -1
\][/tex]
### Summary:
The solutions to the equation [tex]\( |2u - 5| = 7 \)[/tex] are:
[tex]\[
u = 6 \quad \text{and} \quad u = -1
\][/tex]
So the solutions are [tex]\( u = 6, -1 \)[/tex].
