Verified Answer :

  • The absolute value equation \(|x - 7| = 2\) indicates that \(x - 7\) is 2 units away from zero.
  • This results in two linear equations: \(x - 7 = 2\) and \(x - 7 = -2\).
  • Solving the first equation gives \(x = 9\), and solving the second gives \(x = 5\).
  • The solution set is \(\boxed{x = 5, 9}\).

Explanation

  1. Analyzing the Equation

The equation given is \(|x - 7| = 2\). The absolute value of a number represents its distance from zero on the number line. Therefore, the expression \(|x - 7| = 2\) indicates that the value inside the absolute value, \(x - 7\), must be at a distance of 2 from zero. This means that \(x - 7\) can be either 2 or -2.

  1. Setting up the Cases

To solve the equation \(|x - 7| = 2\), we break it down into two separate linear equations based on the definition of the absolute value function:
Case 1: \(x - 7 = 2\)
Case 2: \(x - 7 = -2\)

  1. Solving the Equations

Now we solve each linear equation individually:
For Case 1:
\(x - 7 = 2\)
Add 7 to both sides of the equation:
\(x = 2 + 7\)
\(x = 9\)

For Case 2:
\(x - 7 = -2\)
Add 7 to both sides of the equation:
\(x = -2 + 7\)
\(x = 5\)

  1. Verification and Final Answer

We have found two potential solutions: \(x = 9\) and \(x = 5\).
We can verify these solutions by substituting them back into the original equation:
If \(x = 9\): \(|9 - 7| = |2| = 2\). This is correct.
If \(x = 5\): \(|5 - 7| = |-2| = 2\). This is also correct.
Thus, the solutions to the equation are \(x = 5\) and \(x = 9\).

Examples

Absolute value equations are useful for finding ranges or thresholds. For example, if a machine produces metal rods and the target length is 7 centimeters, with an allowable tolerance of 2 centimeters, you can use the equation \(|x - 7| = 2\) to find the minimum and maximum acceptable lengths of the rods. Here, the acceptable lengths would be 5 cm and 9 cm. Any rod shorter than 5 cm or longer than 9 cm would be outside the tolerance range.

Related Questions