Inverse variation
y varies inversely with x. If y=8 when x=3, find the value of y when x=−6.
y = 8 when x = 3, so k = xy = 24. When x = -6: y = 24/(-6) = -4
Find the horizontal and vertical asymptotes of the graph of f(x) = (2x + 1)/(x - 5).
Vertical asymptote: x = 5; Horizontal asymptote: y = 2
Multiply the following rational expressions and state any restrictions on the variables: (x² - 4)/(x + 1) × (x - 3)/(x² - 2x - 8)
(x - 2)(x - 3)/(x + 1)(x + 2); Restrictions: x ≠ -1, -2, 4
Simplify (x² - 6x + 9)/(x² - 9) - (2x - 6)/(x² - 9) and state any restrictions on the variables.
(x - 1)/(x + 3); Restrictions: x ≠ 3, -3
Solve the equation: x/(x - 4) + 2/(x + 2) = 12/(x² - 2x - 8)
x = 2
It takes Pipe A 8 hours to fill a swimming pool, and it takes Pipe B 12 hours to fill the same pool. If both pipes work together, how many hours will it take to fill the pool?
1/8 + 1/12 = 1/t; t = 4.8 hours or 4 hours 48 minutes
z varies inversely with w. If z=−5 when w=4, find the value of ww when z=10.
z = -5 when w = 4, so k = zw = -20. When z = 10: w = -20/10 = -2
Determine the horizontal and vertical asymptotes of the function g(x) = (3x2 - 2)/(x + 4).
Vertical asymptote: x = -4; Horizontal asymptote: none
Divide the following rational expressions and state any restrictions on the variables: (2x² + 5x - 3)/(x² - 9) ÷ (2x - 1)/(x + 3)
(x + 3)²/(x - 3); Restrictions: x ≠ 3, -3, 1/2
Simplify (x² + 3x - 10)/(x² - 25) + (2x + 6)/(x² - 25) and state restrictions.
(x + 5)/(x - 5); Restrictions: x ≠ 5, -5
Solve: 3/(x - 1) - 2/(x + 3) = 1/(x² + 2x - 3)
x = 4
Sarah can mow her lawn in 3 hours. Her brother Jake can mow the same lawn in 4 hours. How long will it take them to mow the lawn if they work together?
1/3 + 1/4 = 1/t; t = 12/7 hours or about 1.71 hours (1 hour 43 minutes)
The time it takes to complete a job varies inversely with the number of workers. If 6 workers can complete the job in 8 hours, how long will it take 4 workers to complete the same job?
Workers × Time = constant. 6 × 8 = 48. For 4 workers: 4t = 48, so t = 12 hours
Find all asymptotes of h(x) = (x - 4)/(x² - 9)
Vertical asymptotes: x = 3, x = -3; Horizontal asymptote: y =0
Multiply and simplify: (x² - 6x + 9)/(x² + 2x - 15) × (x² - 25)/(x² - 9)
(x - 3)(x - 5)/(x + 5)(x + 3); Restrictions: x ≠ -5, 3, -3
Simplify (2x² - 8)/(x² + 4x + 4) × (x + 2)/(x - 2) and state restrictions.
2(x + 2)/(x + 2) = 2; Restrictions: x ≠ -2, 2
Solve: (x + 1)/(x - 2) = (x - 3)/(x + 4) + 2/(x² + 2x - 8)
x = 0.4
or
x = 2/5
A printing press can complete a job in 6 hours. A second press can complete the same job in 8 hours. After working together for 2 hours, the second press breaks down. How much longer will it take the first press to finish the job alone?
After 2 hours together, 1/3 of job remains. First press alone: 2 more hours
a varies inversely with the square of b. If a=12 when b=2, find the value of a when b=6.
a = 12 when b = 2, so k = ab² = 12(4) = 48. When b = 6: a = 48/36 = 4/3
Determine the asymptotes of f(x) = (2x² + 3x - 1)/(x² + x - 6).
Vertical asymptotes: x = 2, x = -3; Horizontal asymptote: y = 2
Divide and simplify: (3x² + 10x - 8)/(x² + 6x + 8) ÷ (6x² - 5x - 4)/(2x² + 9x + 4)
(x + 4)/(2x + 1); Restrictions: x ≠ -4, -2, 4/3, -1/2
Simplify (x² - 16)/(x² + 8x + 16) + (x + 4)/(x² + 8x + 16) - (2x)/(x² + 8x + 16).
(x - 2)/(x + 4); Restrictions: x ≠ -4
1/(x - 4) + 1/(x + 4) = 8/(x² - 16)
No solution (the algebraic solution x = 4 is extraneous because it makes denominators zero)
The resistance of two resistors connected in parallel is given by 1/R = 1/R₁ + 1/R₂. If the total resistance is 6 ohms and one resistor has a resistance of 10 ohms, find the resistance of the second resistor.
1/6 = 1/10 + 1/R₂; R₂ = 15 ohms