The LCD is а Multiple of Each Denominator - Outline Problem Sets

Think and Discuss: Possible answer: The LCD is a multiple of each denominator. Therefore, each denominator is a factor of the LCD. Possible answer: When you multiply both sides of an equation by a variable expression, you may produce an equation with solutions that make denominators of the original equation equal to 0.   Possible answer: (1) Graph each side of the inequality. (2) Multiply both sides by x and consider two cases: x is positive or x is negative.   Possible answer: It will take Remy about 24 min to mulch the garden when working alone. Exercises 5a. (3 < x \leq 4) 5b. (x = -5) 6a. LCD is positive. 6/(x-2) ≥ -4 6 ≥ -4(x-2) 6 ≥ -4x + 8 4x ≥ 2 x ≥ 1/2 x - 2 > 0 x > 2 Solution in this case is x > 2. 6b. LCD is negative. 6/(x-2) ≥ -4 6 ≤ -4(x-2) 6 ≤ -4x + 8 4x ≤ 2 x ≤ 1/2 x - 2 < 0 x < 2 Solution in this case is x ≤ 1/2. The solution to the inequality is x ≤ 1/2 or x > 2. 7a. LCD is positive. 3/(x+3) > 6 9 > 6x + 18 -9 > 6x x < -3/2 x + 3 > 0 x > -3 The solution in this case is x < -3/2. 7b. LCD is negative. 3/(x+3) > 6 9 < 6x + 18 -9 > 6x x < -3/2 x + 3 < 0 x < -3 The solution in this case is x < -3. The solution to the inequality is x < -3 or x > -3/2. 8. m + 12/m = 7 m^2 + 12 = 7m m^2 - 7m + 12 = 0 (m-3)(m-4) = 0 m = 3 or m = 4 9. k + 1/k = 2 k^2 + 1 = 2k k^2 - 2k + 1 = 0 (k-1)^2 = 0 k = 1 8. -2x/(x+2) + x/3 = 4/(x+2) (-2x/x+2)3(x+2) + (x/3)3(x+2) = (4/x+2)3(x+2) -6x + x(x+2) = 12 x^2 - 4x - 12 = 0 (x-6)(x+2) = 0 x = 6 or x = -2 The only solution is x = 6. The solution x = -2 is extraneous. 9. x/(x-3) + x/2 = 6x/(2x-6) x/(x-3)(2x-6) + x/2(2x-6) = 6x/(2x-6)(2x-6) x^2 - 3x = 6x x^2 - 9x = 0 x(x-9) = 0 x = 0 or x = 9 10. 3/[x(x+1)] - 1 = 3/(x^2+x) 3/[x(x+1)]x(x+1) - 1x(x+1) = 3/(x^2+x)x(x+1) 3 - x(x+1) = 3 x^2 + x = 0 x(x+1) = 0 x = 0 or x = -1 11. 12/(x-4) <= 3 12/(x-4)(x-4) <= 3(x-4) 12 <= 3x - 12 24 <= 3x 8 <= x x - 4 > 0 x > 4 The solution in this case is x > 4. 12. 12/(x-4) <= 3 12/(x-4)(x-4) >= 3(x-4) 12 >= 3x - 12 24 >= 3x 8 >= x x - 4 < 0 x < 4 The solution in this case is x <= 4. The solution to the inequality is x <= 4 or x > 4. 13. 10/(x+8) > 2 10/(x+8)(x+8) > 2(x+8) 10 > 2x + 16 -6 > 2x x < -3 x + 8 < 0 x < -8 No solution in this case. 14. 10/(x+8) < 2 10/(x+8)(x+8) < 2(x+8) 10 < 2x + 16 -6 < 2x x > -3 x + 8 > 0 x > -8 The solution to the inequality is -8 < x < -3. 1/(z-3) = 7 1 = 7(z-3) 1 = 7z - 21 22 = 7z z = 22/7 1/(x-4) + 1/(x+4) = 1 (1/(x-4) + 1/(x+4))(x-4)(x+4) = 1(x-4)(x+4) (x+4) + (x-4) = x^2 - 16 2x = x^2 - 16 x^2 - 2x - 16 = 0 (x-1)(x+4) = 0 x = 1 or x = -4 14/z = 9 - z 14 = 9z - z^2 z^2 - 9z + 14 = 0 (z-2)(z-7) = 0 z = 2 or z = 7 x/(x-4) = 4/(x-4) x = 4 (2x)/(x+3) - 12/(x^2-9) = 3/(x-3) (2x/(x+3) - 12/((x-3)(x+3)) = 3/(x-3)) (x-3)(x+3) 2x(x-3) - 12 = 3(x+3) 2x^2 - 6x - 12 = 3x + 9 2x^2 - 9x - 21 = 0 (2x+3)(x-7) = 0 x = -3/2 or x = 7 The solution x = -3/2 is extraneous. The only solution is x = 7. 3x/(x+1) = (2x-1)/(x+1) 3x = 2x - 1 x = -1 The solution x = -1 is extraneous. Therefore, there is no solution. 1/(x-2) + 1/(x-1) = 2/(x^2-3x+2) (1/(x-2) + 1/(x-1)) (x-2)(x-1) = 2/(x^2-3x+2) (x-2)(x-1) (x-1) + (x-2) = 2 2x - 3 = 2 2x = 5 x = 5/2 The solution x = 5/2 is extraneous. Therefore, there is no solution. 12/(x-4) <= 3 12/(x-4)(x-4) <= 3(x-4) 12 <= 3x - 12 24 <= 3x 8 <= x x - 4 > 0 x > 4 The solution in this case is x > 4. 12/(x-4) <= 3 12/(x-4)(x-4) >= 3(x-4) 12 >= 3x - 12 24 >= 3x 8 >= x x - 4 < 0 x < 4 The solution in this case is x <= 4. The solution to the inequality is x <= 4 or x > 4. 10/(x+8) > 2 10/(x+8)(x+8) > 2(x+8) 10 > 2x + 16 -6 > 2x x < -3 x + 8 < 0 x < -8 No solution in this case. 10/(x+8) < 2 10/(x+8)(x+8) < 2(x+8) 10 < 2x + 16 -6 < 2x x > -3 x + 8 > 0 x > -8 The solution to the inequality is -8 < x < -3.