Linear Inequalities

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Step 1: Subtract 3 from both sides → 5x < 15 Step 2: Divide both sides by 5 → x < 3 Answer: x < 3, or x ∈ (-∞, 3) Key point: When dividing by a positive number, the inequality sign stays the same.

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Step 1: Subtract 7 from both sides → -3x ≥ -6 Step 2: Divide both sides by -3. Since we divide by a NEGATIVE number, REVERSE the inequality sign → x ≤ 2 Answer: x ≤ 2, or x ∈ (-∞, 2] Critical rule: A…

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You reverse the inequality sign ONLY when: 1. Multiplying both sides by a negative number 2. Dividing both sides by a negative number Example: -2x > 8 Step 1: Divide by -2 (negative!) → x < -4 (sign …

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Step 1: Subtract 2x from both sides → 5x - 5 > 10 Step 2: Add 5 to both sides → 5x > 15 Step 3: Divide by 5 (positive number) → x > 3 Answer: x > 3, or x ∈ (3, ∞) Graphical representation: Open circl…

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Step 1: Multiply both sides by 6 (LCM of 3 and 2) → 2(5x - 2) ≤ 3(x + 4) Step 2: Expand → 10x - 4 ≤ 3x + 12 Step 3: Subtract 3x from both sides → 7x - 4 ≤ 12 Step 4: Add 4 to both sides → 7x ≤ 16 Step…

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This means: 2 ≤ 3x - 1 AND 3x - 1 < 8 For the left part: 2 ≤ 3x - 1 Step 1: Add 1 to both sides → 3 ≤ 3x Step 2: Divide by 3 → 1 ≤ x (or x ≥ 1) For the right part: 3x - 1 < 8 Step 1: Add 1 to both s…

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Step 1: Multiply all parts by 4 → -20 ≤ 3 - 2x ≤ 4 For the left part: -20 ≤ 3 - 2x Step 2a: Subtract 3 from both sides → -23 ≤ -2x Step 3a: Divide by -2 (negative!), reverse signs → 23/2 ≥ x (or x ≤ …

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'x < 5' (strict inequality): - Represented by an OPEN circle at 5 - Does NOT include 5 itself - All numbers less than 5 - Interval notation: (-∞, 5) 'x ≤ 5' (slack inequality): - Represented by a CLO…