Solve the absolute value equation. 4- 10x, =5 Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is (Use a comma to separate answers as needed.) B. The solution set is all real numbers. C. The solution is the empty set.
The Correct Answer and Explanation is:
We are given the absolute value equation:
|4 – 10x| = 5
Step 1: Understand the definition of absolute value
The absolute value of a number is its distance from 0 on the number line, so:
|A| = B means A = B or A = -B, as long as B is positive.
So, from |4 – 10x| = 5, we write two equations:
1) 4 – 10x = 5
2) 4 – 10x = -5
Step 2: Solve each equation
Equation 1: 4 – 10x = 5
Subtract 4 from both sides:
-10x = 1
Divide both sides by -10:
x = -1/10
Equation 2: 4 – 10x = -5
Subtract 4 from both sides:
-10x = -9
Divide both sides by -10:
x = 9/10
Step 3: State the solution set
We found two values that make the original equation true:
x = -1/10 and x = 9/10
So the solution set is:
A. The solution set is -1/10, 9/10
Explanation
This problem involves solving an absolute value equation. The absolute value expression |4 – 10x| represents how far the quantity 4 – 10x is from zero, regardless of direction. The equation says this distance equals 5, which means the inside expression could be either 5 or -5, because both are 5 units from zero.
To solve the equation, we break it into two simpler linear equations:
- One assumes the quantity inside the absolute value is positive: 4 – 10x = 5
- The other assumes it’s negative: 4 – 10x = -5
Solving both equations gives us two possible x-values. For the first, subtract 4 from both sides to get -10x = 1, then divide by -10 to find x = -1/10. For the second, subtract 4 to get -10x = -9, then divide by -10 to find x = 9/10.
Since both solutions are valid, the solution set consists of these two values. Therefore, the correct answer is:
A. The solution set is -1/10, 9/10
This process demonstrates a basic principle in algebra: absolute value equations usually lead to two separate equations, which must be solved individually. Always check for extraneous solutions in more complex cases, but here, both solutions work.
