1. Solve each equation and classify it. (a) Solve $$\frac{1}{2}x + 6 = \frac{3}{4}x - 2 - \frac{1}{4}x + 8$$ Combine like terms on the right: $$\frac{3}{4}x - \frac{1}{4}x = \frac{2}{4}x = \frac{1}{2}x$$ So equation becomes: $$\frac{1}{2}x + 6 = \frac{1}{2}x + 6$$ Subtract $$\frac{1}{2}x$$ from both sides: $$\cancel{\frac{1}{2}x} + 6 = \cancel{\frac{1}{2}x} + 6$$ $$6 = 6$$ This is true for all $$x$$, so it is an identity. (b) Solve $$5x - (3 + x) + 2x + 4 = 7x + 6$$ Distribute minus: $$5x - 3 - x + 2x + 4 = 7x + 6$$ Combine like terms left: $$ (5x - x + 2x) + (-3 + 4) = 7x + 6$$ $$6x + 1 = 7x + 6$$ Subtract $$6x$$ from both sides: $$\cancel{6x} + 1 = x + 6$$ Subtract 6 from both sides: $$1 - 6 = x + \cancel{6} - 6$$ $$-5 = x$$ This is a conditional equation with solution $$x = -5$$. (c) Solve $$\frac{1}{6}(x + 3) + \frac{1}{4}(x - 2) = \frac{5}{12}x + \frac{1}{2}$$ Distribute: $$\frac{1}{6}x + \frac{3}{6} + \frac{1}{4}x - \frac{2}{4} = \frac{5}{12}x + \frac{1}{2}$$ Simplify constants: $$\frac{1}{6}x + \frac{1}{2} + \frac{1}{4}x - \frac{1}{2} = \frac{5}{12}x + \frac{1}{2}$$ $$\frac{1}{6}x + \frac{1}{4}x + (\frac{1}{2} - \frac{1}{2}) = \frac{5}{12}x + \frac{1}{2}$$ $$\frac{1}{6}x + \frac{1}{4}x = \frac{5}{12}x + \frac{1}{2}$$ Find common denominator for left terms: $$\frac{2}{12}x + \frac{3}{12}x = \frac{5}{12}x + \frac{1}{2}$$ $$\frac{5}{12}x = \frac{5}{12}x + \frac{1}{2}$$ Subtract $$\frac{5}{12}x$$ from both sides: $$\cancel{\frac{5}{12}x} = \cancel{\frac{5}{12}x} + \frac{1}{2}$$ $$0 = \frac{1}{2}$$ This is false, so it is a contradiction. (d) Solve $$\frac{1}{5}(x + 10) + \frac{1}{10}(x - 5) = \frac{3}{10}x + \frac{1}{2}$$ Distribute: $$\frac{1}{5}x + 2 + \frac{1}{10}x - \frac{1}{2} = \frac{3}{10}x + \frac{1}{2}$$ Combine constants: $$\frac{1}{5}x + \frac{1}{10}x + (2 - \frac{1}{2}) = \frac{3}{10}x + \frac{1}{2}$$ $$\frac{1}{5}x + \frac{1}{10}x + \frac{3}{2} = \frac{3}{10}x + \frac{1}{2}$$ Convert $$\frac{1}{5}x$$ to $$\frac{2}{10}x$$: $$\frac{2}{10}x + \frac{1}{10}x + \frac{3}{2} = \frac{3}{10}x + \frac{1}{2}$$ $$\frac{3}{10}x + \frac{3}{2} = \frac{3}{10}x + \frac{1}{2}$$ Subtract $$\frac{3}{10}x$$ from both sides: $$\cancel{\frac{3}{10}x} + \frac{3}{2} = \cancel{\frac{3}{10}x} + \frac{1}{2}$$ $$\frac{3}{2} = \frac{1}{2}$$ False, so contradiction. 2. Solve for specified variable. (a) Solve $$C = \frac{5}{9}(F - 32)$$ for $$F$$. Multiply both sides by 9: $$9C = 5(F - 32)$$ Divide both sides by 5: $$\frac{9C}{5} = F - 32$$ Add 32 to both sides: $$F = \frac{9C}{5} + 32$$ (b) Solve $$2x - 4y = 10$$ for $$y$$. Subtract $$2x$$ from both sides: $$-4y = 10 - 2x$$ Divide both sides by $$-4$$: $$y = \frac{10 - 2x}{-4} = \frac{\cancel{2}(5 - x)}{\cancel{-4}(-2)} = -\frac{5 - x}{2} = \frac{x - 5}{2}$$ 3. Percent of items not returned. Total items = 500, returned = 125. Not returned = $$500 - 125 = 375$$ Percent not returned: $$\frac{375}{500} \times 100 = 75\%$$ 4. Salt and water mixture. Salt = 28% of 50 L: $$0.28 \times 50 = 14\text{ L salt}$$ Water = $$50 - 14 = 36\text{ L water}$$ 5. Time for cyclist. Distance = 150 miles, speed = 30.5 mph. Time = $$\frac{150}{30.5} \approx 4.918$$ hours (to nearest thousandth). 6. Percent increase in population. Initial = 1.5 million, final = 1.8 million. Increase = $$1.8 - 1.5 = 0.3$$ million. Percent increase: $$\frac{0.3}{1.5} \times 100 = 20.0\%$$ 7. Simple interest rate. Principal $$P = 15000$$, Interest $$I = 2250$$, Time $$t = 5$$ years. Formula: $$I = P \times r \times t$$ Solve for $$r$$: $$r = \frac{I}{P \times t} = \frac{2250}{15000 \times 5} = 0.03 = 3\%$$ 8. Body surface area and child dose. Weight = 35 lb. Formula for body surface area (BSA): $$BSA = \sqrt{\frac{weight \times height}{3600}}$$ Height not given, so assume standard formula for weight only: Use approximate formula: $$BSA = 0.1 \times weight^{0.67}$$ (approximation) Calculate: $$BSA \approx 0.1 \times 35^{0.67} \approx 0.1 \times 11.5 = 1.15\, \text{m}^2$$ Child dose proportional to BSA: $$\frac{child\ dose}{600} = \frac{1.15}{1.73}$$ $$child\ dose = 600 \times \frac{1.15}{1.73} \approx 399$$ mg Final answers: (a) Identity (b) Conditional, $$x = -5$$ (c) Contradiction (d) Contradiction 2(a) $$F = \frac{9C}{5} + 32$$ 2(b) $$y = \frac{x - 5}{2}$$ 3. 75% 4. 14 L salt, 36 L water 5. 4.918 hours 6. 20.0% 7. 3% 8. BSA approx 1.15 m², child dose approx 399 mg