Step 1: Cross-multiply: x · x = 3 · 4 x² = 12

Step 2: Solve for x: x = ±√12 = ±2√3

Step 3: Check both solutions: x = 2√3: (2√3)/3 = 4/(2√3) = 4√3/6 = 2√3/3 ✓ x = -2√3: (-2√3)/3 = 4/(-2√3) = -4√3/6 = -2√3/3 ✓

Answer: x = ±2√3

Cross multiply: $2x = 3 \cdot 5$ $2x = 15$ $x = \frac{15}{2}$

Check: $\frac{15/2}{3} = \frac{15}{6} = \frac{5}{2}$ ✓

Answer: $x = \frac{15}{2}$ or $7.5$

Step 1: Cross-multiply: 5(x + 1) = 3(x - 2)

Step 2: Expand both sides: 5x + 5 = 3x - 6

Step 3: Solve for x: 5x - 3x = -6 - 5 2x = -11 x = -11/2

Step 4: Check for restrictions: x ≠ 2, -1 (would make denominators zero) x = -11/2 doesn't violate restrictions ✓

Step 5: Verify solution: 5/(-11/2 - 2) = 5/(-15/2) = -10/15 = -2/3 3/(-11/2 + 1) = 3/(-9/2) = -6/9 = -2/3 ✓

Answer: x = -11/2

LCD: $2x(x + 2)$

Multiply both sides by LCD: $2x(x + 2) \cdot \frac{1}{x} + 2x(x + 2) \cdot \frac{1}{x+2} = 2x(x + 2) \cdot \frac{1}{2}$

$2(x + 2) + 2x = x(x + 2)$ $2x + 4 + 2x = x^2 + 2x$ $4x + 4 = x^2 + 2x$ $0 = x^2 - 2x - 4$

Use quadratic formula: $a = 1, b = -2, c = -4$ $x = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = \frac{2 \pm 2\sqrt{5}}{2} = 1 \pm \sqrt{5}$

Answer: $x = 1 + \sqrt{5}$ or $x = 1 - \sqrt{5}$

Step 1: Find the LCD: LCD = 2x(x - 3)

Step 2: Multiply every term by LCD: 2x(x - 3) · (2/x) + 2x(x - 3) · [1/(x - 3)] = 2x(x - 3) · (1/2)

Step 3: Simplify each term: 4(x - 3) + 2x = x(x - 3) 4x - 12 + 2x = x² - 3x 6x - 12 = x² - 3x

Step 4: Rearrange to standard form: 0 = x² - 3x - 6x + 12 0 = x² - 9x + 12

Step 5: Solve using quadratic formula: x = [9 ± √(81 - 48)]/2 x = [9 ± √33]/2

Step 6: Check restrictions: x ≠ 0, 3 Both solutions are valid

Answer: x = (9 + √33)/2 or x = (9 - √33)/2

Step 1: Factor x² - 1: x² - 1 = (x + 1)(x - 1)

Step 2: Find LCD: LCD = (x + 1)(x - 1)

Step 3: Multiply every term by LCD: (x + 1)(x - 1) · [x/(x - 1)] - (x + 1)(x - 1) · [2/(x + 1)] = (x + 1)(x - 1) · [4/((x + 1)(x - 1))]

Step 4: Simplify: x(x + 1) - 2(x - 1) = 4

Step 5: Expand: x² + x - 2x + 2 = 4 x² - x + 2 = 4

Step 6: Solve: x² - x - 2 = 0 (x - 2)(x + 1) = 0 x = 2 or x = -1

Step 7: Check restrictions: x ≠ 1, -1 x = -1 is EXTRANEOUS (makes denominator 0) x = 2 is valid ✓

Answer: x = 2

LCD: $x - 2$

Multiply both sides: $6 = x + (x - 2)$ $6 = x + x - 2$ $6 = 2x - 2$ $8 = 2x$ $x = 4$

Check: Does $x = 4$ make any denominator zero? $x - 2 = 4 - 2 = 2 \neq 0$ ✓

Verify: $\frac{6}{2} = \frac{4}{2} + 1$ → $3 = 2 + 1$ ✓

Answer: $x = 4$

Step 1: Set up rates: John's rate: 1/6 of room per hour Mary's rate: 1/4 of room per hour Combined rate: 1/6 + 1/4

Step 2: Find LCD and add rates: LCD = 12 1/6 = 2/12 1/4 = 3/12 Combined: 2/12 + 3/12 = 5/12 of room per hour

Step 3: Set up equation: If t = time to paint together (5/12)t = 1 room

Step 4: Solve for t: t = 1 ÷ (5/12) t = 1 · (12/5) t = 12/5 t = 2.4 hours

Step 5: Convert to hours and minutes: 2.4 hours = 2 hours + 0.4(60 minutes) = 2 hours 24 minutes

Step 6: Verify: In 2.4 hours: John paints: 2.4/6 = 0.4 or 2/5 Mary paints: 2.4/4 = 0.6 or 3/5 Total: 2/5 + 3/5 = 5/5 = 1 room ✓

Answer: 12/5 hours or 2 hours 24 minutes