The decision and its conditions
Absolute value encodes a nonnegative distance, but the expression inside the bars can be nonlinear. A positive right side permits two equations; a zero right side collapses them into one condition. A greater-than distance inequality describes the exterior of an interval rather than its interior. When a problem also restricts the allowed inputs, intersect that domain with the distance condition. Comparing two absolute values can be done by equal or opposite signed inner expressions, provided all solutions are checked.
Worked example
What is the sum of the positive solutions of |x²−10|=6? The two cases are x²=16 and x²=4. The positive solutions are 4 and 2. Their sum is 6; negative roots exist but are not requested.
A contrasting decision
For integer x with -4≤x≤5, how many satisfy |x−1|≥2? All allowed integers number 10. The only failures are 0, 1, and 2, where distance is less than two. Subtract three failures to get 7; both distance-two boundaries are included.
Diagnose a competing method
The first example’s tempting answer 0 follows this mistake: Sum all four signed roots instead of only the positive ones. Split the absolute-value equation before taking square roots, and keep only the requested positive roots.
Transfer before revealing
Find the real solution of |x−4|=|x−7|. Pause and identify the original restrictions. Solution: The point is equally distant from the distinct locations 4 and 7. The opposite-sign case gives 2x=11; the same-sign case would require 4=7. Therefore x=11/2, their midpoint.
Check the boundary of the method
When a problem also restricts the allowed inputs, intersect that domain with the distance condition. Count the finite allowed domain, then remove the open interior that fails the exterior condition. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
What is the sum of the positive solutions of |x²−10|=6?
- The two cases are x²=16 and x²=4.
- The positive solutions are 4 and 2.
- Their sum is 6; negative roots exist but are not requested.
Compare the other answer choices
Sum all four signed roots instead of only the positive ones.
Keep only the plus case inside the absolute value.
Add squared values instead of their positive roots.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overview