Question

define and explain reduced inequalities?

Answer 1

Could you say which context you mean (high‑school algebra, systems of inequalities, linear programming, or something else)? Different fields use the phrase differently. Meanwhile, here are the two most common meanings and helpful rules/examples.

1) “Reduced” as “simplified” (algebra / solving inequalities)
- Definition: an inequality is reduced when it’s been simplified to a standard form (typically isolating the variable), e.g. x < 3 or ax + b ≥ 0.
- How to reduce an inequality:
- Combine like terms and move all terms with the variable to one side and constants to the other.
- You may add or subtract the same number from both sides without changing the inequality.
- If you multiply or divide both sides by a positive number the inequality direction stays the same; if you multiply or divide by a negative number the direction reverses.
- Never divide by zero.
- Examples:
- Linear: 3x − 7 > 2x + 1 ⇒ x > 8.
- Multiplying by negative: −2x + 5 ≤ 3 ⇒ −2x ≤ −2 ⇒ x ≥ 1 (note sign flip).
- Absolute value: |x − 4| < 3 ⇒ 1 < x < 7 (or 4 − 3 < x < 4 + 3).
- Quadratic: x^2 − 5x + 6 < 0 ⇒ (x − 2)(x − 3) < 0 ⇒ solution is 2 < x < 3 (use sign chart).
- Rational: (x − 2)/(x + 3) > 0 ⇒ critical points −3 (undefined) and 2 (zero); test intervals ⇒ solution (−∞, −3) ∪ (2, ∞).

2) “Reduced” as “reduced system of inequalities” (removing redundancy)
- In systems (or linear programming), a reduced system has redundant inequalities removed (an inequality is redundant if it’s implied by the others).
- How to test redundancy: temporarily remove an inequality and see whether the feasible region changes (in LP this can be done by solving an optimization or checking if that inequality is implied by others).
- Benefit: smaller system is easier to analyze/solve.

Other useful tips
- For rational or polynomial inequalities, use sign charts: list critical points (zeros and undefined points), mark intervals, test a point in each interval to determine sign.
- Keep track of strict (>,<) vs nonstrict (≥, ≤) at endpoints: equalities allow endpoints; strict do not.
- Pay attention to domain restrictions (e.g., denominators and even roots).

If you meant a specific usage (a formal definition in optimization theory, a particular textbook term, or a worked example), tell me which and I’ll give a focused explanation or walk through examples step by step.