Homework 3

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. A student tries to simplify \[4 - \dfrac{x+3}{x + 2} = \dfrac{4 - x+3}{x+2}\] What did the student forget?
  2. A friend you know tries to simplify \[\dfrac{x^{-1} + y^{-1}}{4} = \dfrac{1}{4xy}\] Why is this incorrect?
  3. When simplifying fractional expressions, what five goals are met to be considered simplified?

    4. For problems 4-6, find the LCD.

  4. $\dfrac{1}{x}$ and $\dfrac{1}{x^2 - 1}$
  5. $\dfrac{1}{x^2 - 2x + 1}$ and $\dfrac{1}{x^2 - 1}$
  6. $\dfrac{1}{(x+2)^3(x^2+1)}, \dfrac{1}{3x}$ and $\dfrac{1}{x + 2}$

    7. For problems 7-22, fully simplify each expression.

  7. $\dfrac{2(x+h) - 3 - (2x - 3)}{h}$
  8. $\dfrac{(x+h)^2 - x^2}{h}$
  9. $\dfrac{(x+h)^2 - (x + h) - (x^2 - x)}{h}$
  10. $\dfrac{x^2 + 7x + 12}{x^2 + 3x + 2} \cdot \dfrac{x^2 + 5x + 6}{x^2 + 6x + 9}$
  11. $\dfrac{2x^2 - x - 6}{4x^2 - 7x + 3} \div \dfrac{2x^2 + 5x + 3}{x^2 - 4x + 3}$
  12. $2 - \dfrac{x+1}{x-3}$
  13. $x^{-1} + \dfrac{1}{x - 1}$
  14. $\dfrac{x}{x^2 + 2x + 1} - \dfrac{1}{x + 1}$
  15. $\dfrac{x - 2}{x^2 + 5x + 6} - \dfrac{x + 2}{x^2 + 4x + 3}$
  16. $\dfrac{1}{x + 1} - \dfrac{1}{x - 1}$
  17. $\dfrac{1 + \dfrac{1}{x}}{\dfrac{1}{x} - 2}$
  18. $\dfrac{x}{x + 3} - \dfrac{18}{x^2 - 9}$
  19. $\dfrac{1}{x-1} + \dfrac{1}{x} - \dfrac{1}{x+1}$
  20. $\dfrac{\dfrac{1}{1 + h} - 1}{h}$
  21. $\dfrac{x^{-1} + y^{-1}}{4} \ \ $
    Hint Use definition of negative exponent. This is a compound fraction.
  22. $\dfrac{\dfrac{x-1}{x+1}\cdot \dfrac{x^2 - 1}{x}}{x - 1}$

  23. Rationalize the denominator and simplify: $\dfrac{2}{\sqrt{2}}$
  24. Rationalize the denominator and simplify: $\dfrac{h}{\sqrt{x+h} - \sqrt{x}}$
  25. Rationalize the numerator and simplify: $\dfrac{\sqrt{1 + h} - 1}{h}$
  26. Show why $x = 2$ is a solution to the equation \[\dfrac{1}{x} - \dfrac{1}{x-4} = 1\]
  27. Isolate the given variable in the following equations:
    1. $4x + 2 = 6x - w, \ $ for $x$
    2. $\dfrac{x + 1}{x - 3} = 1, \ $ for $x$
    3. $4xy - w(2xz - 3yz) + 3 = -4x - y, \ $ for $x$
    4. $3x^2 - 2(y' + xy') - 3y^2y' = 4, \ $ for $y'$
  28. Find all solutions for the following equations that are real numbers.
    1. $3x + 4 = 7$
    2. $2x + 3 = 7 - 3x$
    3. $2x^2 - 5x = -2$
    4. $2x^2 = 8$
    5. $4x^2 - x = 0$
    6. $x^2 - 2x = -1$
  29. Solve the following equations. Remember to check your work if necessary!
    1. $\dfrac{1}{x} = \dfrac{4}{3x} + 1$
    2. $\dfrac{1}{x}-\dfrac{2}{3\left(x-3\right)}=-\dfrac{4}{x^{2}-9}$
    3. $\dfrac{\dfrac{4}{x^2} - 1}{x} = 0$
    4. $\dfrac{1}{x-1} - \dfrac{2}{x^2} = 0$
    5. $\sqrt{8x - 1} = 3$
    6. $\sqrt{2x + 1} + 1 = x$