Overview
This lesson covers how to solve linear inequalities, graph their solutions on number lines, and express solutions using interval notation.
Solving Basic Inequalities
- Solving an inequality is similar to solving an equation; treat the inequality sign as an equal sign during calculations.
- Example: (2x + 3 > 7); subtract 3 from both sides, then divide by 2 to find (x > 2).
- For inequalities with ">", use an open circle on the number line and shade to the right (greater values).
Graphing and Interval Notation
- Open circles represent strict inequalities (>, <); closed circles (shaded) represent inclusive inequalities (≥, ≤).
- Interval notation: use parentheses for open ends (e.g., (2, ∞)), brackets for closed ends (e.g., [−∞, 6]).
Solving with Fractions and Mixed Signs
- To eliminate fractions, multiply both sides by the denominator.
- Example: (\frac{1}{3}x + 4 \leq 6); subtract 4, then multiply by 3 to get (x \leq 6).
- When graphing (x \leq 6), use a closed circle at 6 and shade to the left.
Multiplying/Dividing by Negatives
- When dividing or multiplying both sides of an inequality by a negative, reverse the inequality sign.
- Example: (-3x > -9); divide by −3 to get (x < 3).
Compound and Double Inequalities
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Solve each part of a compound inequality and graph solutions accordingly.
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Example: (2x - 1 > 7) or (-3x + 2 \geq -1) yields (x > 4) or (x \leq 1).
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Combine intervals using union symbol: (−∞, 1] ∪ (4, ∞).
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For double inequalities, solve all sides simultaneously.
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Example: (−12 < 7x − 5 \leq 9) leads to (−1 < x \leq 2).
Key Terms & Definitions
- Inequality — A mathematical statement indicating one value is larger/smaller than another.
- Open Circle — A hollow dot on a number line showing a value is not included (>, <).
- Closed Circle — A solid dot on a number line showing a value is included (≥, ≤).
- Interval Notation — A format for describing solution sets, using brackets [ ] for inclusive and parentheses ( ) for exclusive endpoints.
- Compound Inequality — An inequality with more than one comparison (e.g., ( a < x \leq b )).
- Union ( ∪ ) — Indicates combined solution sets in interval notation.
Action Items / Next Steps
- Practice solving and graphing additional inequalities.
- Be careful to flip the inequality sign when multiplying/dividing by a negative.
- Review interval notation and graphing conventions.