Keep Both Interval Endpoints Consistent
A compound inequality restricts one expression between two bounds.
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 5≤2x+1<11. Candidate answers: (A) 2<x≤5 (B) 2≤x<5 (C) 2≤x≤5 (D) 4≤x<10.
- Independent check inputs: Solve 7≤2x+1<13. Candidate answers: (A) 3<x≤6 (B) 3≤x<6 (C) 3≤x≤6 (D) 6≤x<12.
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. A compound inequality restricts one expression between two bounds. Apply the same balanced operation to all three parts, retaining which endpoint is included and which is excluded.
Here is the method on the supplied teaching case. Subtract 1 from every part, then divide every part by positive 2. The left endpoint stays inclusive and the right stays exclusive.
An inclusive left boundary and strict right boundary need separate symbols. Draw closed and open endpoints if useful; multiplying by a negative would reverse both comparisons and require reordered bounds.
- 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 5≤2x+1<11. Candidate answers: (A) 2<x≤5 (B) 2≤x<5 (C) 2≤x≤5 (D) 4≤x<10. Solution: Subtract 1 from every part, then divide every part by positive 2. The left endpoint stays inclusive and the right stays exclusive. The final answer is 2≤x<5. An inclusive left boundary and strict right boundary need separate symbols. Draw closed and open endpoints if useful; multiplying by a negative would reverse both comparisons and require reordered bounds.
Check your reasoning
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Helpful tips
- A compound inequality restricts one expression between two bounds.
- An inclusive left boundary and strict right boundary need separate symbols.
See the answer and explanation
3≤x<6
Subtract 1 from every part, then divide every part by positive 2. The left endpoint stays inclusive and the right stays exclusive.
Take it further
Changed-input task: Solve 19≤2x+1<25. Candidate answers: (A) 18≤x<24 (B) 9<x≤12 (C) 9≤x<12 (D) 9≤x≤12. Show the mathematical reason, not only the selected answer. Expected result: 9≤x<12. Subtract 1 from every part, then divide every part by positive 2. The left endpoint stays inclusive and the right stays exclusive.
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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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