Sure, let's solve the equation:
[tex]\[ |v-7| + 11 = 21 \][/tex]
Step-by-step, we'll break this problem down.
### Step 1: Isolate the Absolute Value Expression
First, subtract 11 from both sides of the equation to isolate the absolute value expression:
[tex]\[ |v-7| + 11 - 11 = 21 - 11 \][/tex]
[tex]\[ |v-7| = 10 \][/tex]
### Step 2: Solve the Absolute Value Equation
Next, recall that the equation [tex]\(|A| = B\)[/tex] has two solutions: [tex]\(A = B\)[/tex] and [tex]\(A = -B\)[/tex]. Applying this to our equation:
[tex]\[ v - 7 = 10 \quad \text{or} \quad v - 7 = -10 \][/tex]
### Step 3: Solve Each Case Separately
#### Case 1: [tex]\( v - 7 = 10 \)[/tex]
Add 7 to both sides:
[tex]\[ v - 7 + 7 = 10 + 7 \][/tex]
[tex]\[ v = 17 \][/tex]
#### Case 2: [tex]\( v - 7 = -10 \)[/tex]
Add 7 to both sides:
[tex]\[ v - 7 + 7 = -10 + 7 \][/tex]
[tex]\[ v = -3 \][/tex]
### Final Solution
The solutions to the equation [tex]\( |v-7| + 11 = 21 \)[/tex] are:
[tex]\[ v = 17 \quad \text{and} \quad v = -3 \][/tex]
So, the values of [tex]\( v \)[/tex] that satisfy the given equation are [tex]\( \boxed{17 \text{ and } -3} \)[/tex].
