Linear Inequalities

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An inequality is formed when two real numbers or two algebraic expressions are related by the symbols '<' (less than), '>' (greater than), '≤' (less than or equal to), or '≥' (greater than or equal to

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Strict inequalities use '<' or '>' symbols (e.g., x < 5, 2x + 3 > 7). Slack inequalities use '≤' or '≥' symbols (e.g., x ≤ 5, 2x + 3 ≥ 7). Strict inequalities do not include the boundary value, while

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A linear inequality in one variable x (where a ≠ 0) has the general forms: ax + b < 0, ax + b > 0, ax + b ≤ 0, or ax + b ≥ 0. The variable appears with degree 1 only. Example: 3x - 5 < 7

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A linear inequality in two variables x and y (where a ≠ 0, b ≠ 0) has the general forms: ax + by < c, ax + by > c, ax + by ≤ c, or ax + by ≥ c. Both variables appear with degree 1 only. Example: 2x +

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Rule 1: Equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of inequality. Example: If x + 3 > 5, then x + 3 - 3 > 5 - 3, so x > 2.

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Rule 2: Both sides of an inequality can be multiplied (or divided) by the same positive number without changing the inequality sign. When both sides are multiplied or divided by a negative number, the

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When we multiply or divide both sides by a negative number, the relative order of numbers changes. For example: 3 > 2, but when multiplied by -1, we get -3 < -2. The inequality direction reverses to m

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3x - 5 < 7 3x < 7 + 5 (adding 5 to both sides) 3x < 12 x < 4 (dividing both sides by 3) Solution set: x ∈ (-∞, 4)