A reliable method
For a nonnegative radius r, solve x − a = r and x − a = −r. For a less-than distance inequality, write a compound inequality around the center.
The common trap
An equation with a negative value on the other side of an absolute value has no real solution. Splitting into two cases without checking that sign creates false answers.
Check and transfer
Draw the center and boundary points on a number line. Decide whether endpoints belong to the solution set before counting integer solutions or reporting the interval.
Worked comparison: identify the structure
The solutions of |x−2|=5 are −3 and 7, five units on either side of the center. The larger solution is 7; reporting 5 gives the distance instead of an input. Both candidates can be checked in the original bars.
Transfer and explain
For |x−2|<3, the interval is −1<x<5. Its integer inputs are 0,1,2,3,4: five values. Interval length six is not the count of strictly interior integers.
What is the larger solution of |x − 3| = 6?
- The solutions lie the stated distance to either side of the center.
- The larger solution is the center plus the nonnegative distance.
- The requested value is 9. Substitute or reverse the operation to check this result.
Compare the other answer choices
Report the center rather than an endpoint.
Report the radius alone instead of adding it to the center.
Negate the larger solution by moving the center with the wrong sign.
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 overviewSiyavula Grade 10 foundation reference