A linear inequality describes a range of values rather than a single solution. SAT questions may ask you to solve the inequality, interpret a boundary, identify an integer solution, or model a maximum or minimum.
Essential concepts
\[x<a,;xle a,;x>a,;xge a\]
Use an open endpoint for strict inequalities and a closed endpoint when equality is included.
SAT strategy and application
Solve with a negative coefficient
Solve \(-3x+5ge14\).
- Isolate the variable term
\(-3xge9\).
- Divide and reverse
\(xle-3\).
Solution set: \(xle-3\).
Mistakes and traps
Check your understanding
Solve \(2x-7<5\).
- \(x<6\)
- \(x>6\)
- \(x< -1\)
- \(xle6\)
Show answer and explanation
Answer: \(x<6\)
Add 7 to get \(2x<12\), then divide by positive 2. The strict symbol remains unchanged.
Key takeaways
What to remember
- Interpret an inequality as a complete set of allowable values.
- Reverse the symbol only when multiplying or dividing by a negative.
- Distinguish strict boundaries from inclusive boundaries.
- Apply integer, count, and unit constraints after solving.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.