Solve a Distance Equation in Both Directions
Absolute value measures nonnegative distance from zero.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Solve |x−2|=3. Candidate answers: (A) No solution because an absolute value is positive (B) x=-1 or x=5 (C) x=1 only (D) x=5 only.
- Independent check inputs: Solve |x−3|=4. Candidate answers: (A) No solution because an absolute value is positive (B) x=-1 or x=7 (C) x=1 only (D) x=7 only.
Learn the idea
An equation may describe distance, a bounded interval, a parameter-dependent relation or a formula between several quantities. Its solution must respect all stated conditions, including endpoint inclusion and nonzero divisors. Absolute value measures nonnegative distance from zero. An equation |x−a|=b with b positive locates two points at distance b from a, one on each side.
Here is the method on the supplied teaching case. A distance of 3 from 2 permits both directions. Set x−2=3 or x−2=−3; the solutions are 5 and -1.
Do not keep only the positive direction. Solve both signed equations and check the distances; the two-branch rule would change when the right side were zero or negative.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Solve |x−2|=3. Candidate answers: (A) No solution because an absolute value is positive (B) x=-1 or x=5 (C) x=1 only (D) x=5 only. Solution: A distance of 3 from 2 permits both directions. Set x−2=3 or x−2=−3; the solutions are 5 and -1. The final answer is x=-1 or x=5. Do not keep only the positive direction. Solve both signed equations and check the distances; the two-branch rule would change when the right side were zero or negative.
Check your reasoning
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Helpful tips
- Absolute value measures nonnegative distance from zero.
- Do not keep only the positive direction.
See the answer and explanation
x=-1 or x=7
A distance of 4 from 3 permits both directions. Set x−3=4 or x−3=−4; the solutions are 7 and -1.
Take it further
Changed-input task: Solve |x−9|=10. Candidate answers: (A) No solution because an absolute value is positive (B) x=-1 or x=19 (C) x=1 only (D) x=19 only. Show the mathematical reason, not only the selected answer. Expected result: x=-1 or x=19. A distance of 10 from 9 permits both directions. Set x−9=10 or x−9=−10; the solutions are 19 and -1.
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More practice: try new cases
Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.
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Check the subject sources
- OpenStax: Linear inequalities and absolute value inequalities
- OpenStax: Linear equations in one variable
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