|x-k| + |x-10| = 30

To solve the equation |x-k| + |x-10| = 30, we can follow these steps:

Step 1: Identify the absolute value expressions: |x-k| and |x-10|

Step 2: Set up cases for the absolute value expressions: Case 1: x-k is positive or zero Case 2: x-k is negative Case 3: x-10 is positive or zero Case 4: x-10 is negative

Step 3: Solve each case separately: Case 1: x-k is positive or zero In this case, |x-k| simplifies to (x-k) |x-10| remains the same The equation becomes (x-k) + |x-10| = 30 Simplify further: x - k + |x-10| = 30

Case 2: x-k is negative In this case, |x-k| simplifies to -(x-k) or (k-x) |x-10| remains the same The equation becomes (k-x) + |x-10| = 30 Simplify further: k - x + |x-10| = 30

Case 3: x-10 is positive or zero In this case, |x-10| simplifies to (x-10) |x-k| remains the same The equation becomes |x-k| + (x-10) = 30 Simplify further: |x-k| + x - 10 = 30

Case 4: x-10 is negative In this case, |x-10| simplifies to -(x-10) or (10-x) |x-k| remains the same The equation becomes |x-k| + (10-x) = 30 Simplify further: |x-k| - x + 10 = 30

Step 4: Solve each case separately: Solve the equation in Case 1: x - k + |x-10| = 30 Solve the equation in Case 2: k - x + |x-10| = 30 Solve the equation in Case 3: |x-k| + x - 10 = 30 Solve the equation in Case 4: |x-k| - x + 10 = 30

Step 5: Analyze the solutions obtained from each case and determine if they satisfy the original equation.

By following these steps, you can solve the equation |x-k| + |x-10| = 30.