- To solve the equation \(|x| + 2 = 11\), subtract 2 from both sides to isolate the absolute value: \(|x| = 9\).
- Apply the definition of absolute value to set up two cases: \(x = 9\) and \(x = -9\).
- Both values are integers, matching option A.
- The solution set is \(\boxed{9, -9}\).
Explanation
- Isolating the Absolute Value Expression
To find the integer values of \(x\) that satisfy the equation \(|x| + 2 = 11\), we start by isolating the absolute value term. The absolute value \(|x|\) represents the distance of a number \(x\) from \(0\) on the number line. Subtracting 2 from both sides of the equation gives:
\[|x| = 11 - 2\]
\[|x| = 9\]
- Solving for x
The equation \(|x| = 9\) means that the distance from \(0\) to \(x\) is \(9\). There are two numbers that have a distance of 9 from 0:
- \(x = 9\)
- \(x = -9\)
Both of these values are integers. Therefore, the equation \(|x| + 2 = 11\) is satisfied when \(x = 9\) or \(x = -9\). Looking at the options provided, this corresponds to choice A.
Examples
The absolute value function is often used in real-world scenarios to represent distance or differences regardless of direction. For instance, if you are working on a construction project and need to cut a piece of wood that is 11 inches long, but the tolerance for error is 2 inches (meaning the final length can be 9 to 13 inches), you can use absolute value to find the range. If the target length is \(L\), you might set an equation like \(|L - 11| \leq 2\), which helps ensure your cuts are within acceptable limits. Understanding how to solve equations involving absolute values is foundational for working with tolerances and variations in fields like engineering, manufacturing, and data analysis.