To determine how many solutions exist for the equation [tex]\(\left|\frac{1}{2} x + 1\right| = 5\)[/tex], we need to solve the absolute value equation. Here are the detailed steps:
1. Understanding Absolute Value Equations:
The absolute value equation [tex]\(\left|\frac{1}{2} x + 1\right| = 5\)[/tex] means that [tex]\(\frac{1}{2} x + 1\)[/tex] can be either 5 or -5 because the absolute value of a number [tex]\(a\)[/tex] is equal to 5, if [tex]\(a = 5\)[/tex] or [tex]\(a = -5\)[/tex].
2. Setting Up Two Equations:
We break this into two separate linear equations to solve for [tex]\(x\)[/tex].
- First Equation:
[tex]\[\frac{1}{2} x + 1 = 5\][/tex]
- Second Equation:
[tex]\[\frac{1}{2} x + 1 = -5\][/tex]
3. Solving Each Equation Separately:
- First Equation: [tex]\(\frac{1}{2} x + 1 = 5\)[/tex]
1. Subtract 1 from both sides:
[tex]\[\frac{1}{2} x = 4\][/tex]
2. Multiply both sides by 2:
[tex]\[x = 8\][/tex]
- Second Equation: [tex]\(\frac{1}{2} x + 1 = -5\)[/tex]
1. Subtract 1 from both sides:
[tex]\[\frac{1}{2} x = -6\][/tex]
2. Multiply both sides by 2:
[tex]\[x = -12\][/tex]
4. Counting the Number of Solutions:
From the solving steps above, we find two distinct values of [tex]\(x\)[/tex], which are [tex]\(x = 8\)[/tex] and [tex]\(x = -12\)[/tex].
Therefore, there are 2 solutions to the equation [tex]\(\left|\frac{1}{2} x + 1\right| = 5\)[/tex].
Thus, the correct answer is:
C. 2
