Solve the absolute value equation or indicate that the equation has no solution.

5 StartAbsoluteValue 2 x minus 5 EndAbsoluteValue equals 252x−5=25Question content area bottomPart 1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

The solution set is 

StartSet nothing EndSetenter your response here.

​(Simplify your answer. Use a comma to separate answers as​ needed.)

B.

The solution set is the empty set.

To solve the absolute value equation \( 5|2x - 5| = 25 \), we can start by isolating the absolute value expression. Here's the step-by-step process:

### Step 1: Isolate the absolute value expression
Divide both sides of the equation by 5:
\[
|2x - 5| = \frac{25}{5} = 5
\]

### Step 2: Set up the two cases for the absolute value equation
An absolute value equation \( |A| = B \) implies that \( A = B \) or \( A = -B \). So, we have two cases:

1. **Case 1:** \( 2x - 5 = 5 \)
2. **Case 2:** \( 2x - 5 = -5 \)

### Step 3: Solve each case separately

**Case 1: \( 2x - 5 = 5 \)**
\[
2x = 5 + 5 = 10
\]
\[
x = \frac{10}{2} = 5
\]

**Case 2: \( 2x - 5 = -5 \)**
\[
2x = -5 + 5 = 0
\]
\[
x = \frac{0}{2} = 0
\]

### Step 4: Write the solution set
The solutions from both cases are \( x = 5 \) and \( x = 0 \).

So, the solution set is \( \{0, 5\} \).

### Final Answer
**A.** The solution set is \( \{0, 5\} \).

Would you like further details or have any questions about this solution?

### Follow-up Questions:
1. What happens if the absolute value equation has a negative number on the right side?
2. How do you solve absolute value inequalities instead of equations?
3. Can an absolute value equation ever have exactly one solution? When?
4. What is the geometric interpretation of solving an absolute value equation?
5. How does the solution change if the equation was \( |2x - 5| = 0 \)?

**Tip:** When solving absolute value equations, always remember to consider both the positive and negative cases.

Learn how to solve the absolute value equation |5(2x - 5)| = 25 step-by-step. This problem involves isolating the absolute value expression, setting up cases, solving each case separately, and identifying the solution set. Suitable for high school students in Grades 9-12.

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