May 28, 2014
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Transcript

  • 1. Final Exam Review:
  • 2. Final Exam Review: Eliminate the ‘y’ by adding the two equations.Solve for x, plug back in to find y. 3. Three times the larger of two consecutive odd numbers is five less that four times the smaller. Find the numbers. This is from September, you should be able to solve!! A) 8, 10 B) 15, 17 C) 21, 23 D) 11, 13 E) 8, 9 3(x + 2) = 4x – 5; 3x + 6 = 4x – 5; x = 11
  • 3. Simplifying Rational Expressions A Rational Expression as a fraction where the numerator and the denominator are polynomials. Ex. x²-y² (x-y)² To Simplify a rational expression: 1. Factor the numerator & denominator 2. Divide out any common factors We are working with ratios. Fractions are a type of ratio, where the part is compared to the whole. All the rules of fractions still apply, including the impossibility of zero as a denominator.
  • 4. Simplifying Rational Expressions Think about it. If the denominator, (the ‘whole’ part of the fraction) is zero, how can there be a ‘part’ (the numerator). You can’t have a part of nothing. If the denominator is zero, there is no problem to solve, since this is impossible. Therefore, excluded values mean, “we can solve this problem as long as x doesn’t make the denominator zero.”
  • 5. Simplifying Rational Expressions (x - y)(x + y) (x - y)2 (x-y)² is equal to (x-y)(x-y) so we can cancel out one of the (x-y) To simplify we first factor the polynomials, then cancel any common factors if possible. x + y x - y= Simple, yes?
  • 6. Practice 1  3x2 - 4x 2x2 - x Answer 3x – 4 2x - 1 3x2 - 4x x(3x - 4) 2x2 - x x(2x - 1) == Excluded Values: Pay attention and you’ll get it In the above examples, the excluded values are 0, 𝟏 𝟐 . Here’s why... 1. When we cancelled the x’s, we divided by x. Since dividing by zero is undefined, x cannot be zero. 2. Set the denominator equal to zero, and solve for x. You’ll get 1/2. If x is one-half, the denominator is zero and you won’t have a problem to solve in the first place. Thus, x cannot = 1/2
  • 7. Practice 2
  • 8. Once more, simplify & state the excluded values ?
  • 9. Practice 3 What is/are the excluded value(s) in this expression? x ≠ -6; division by zero. x ≠ 6; undefined denominator
  • 10. By now you can see that factoring is an often used method for simplifying rational expressions. Sometimes these factors are inverses (their product = -1) of each other. In this case, you can manipulate one or the other factor to simplify further. **Doing this will change the sign of the resulting fraction. simplify x2 – 6x + 8 (4 – x)(x + 1) (x - 4)(x – 2) (4 – x)(x + 1) (x - 4)(x – 2) (4 – x)(x + 1) (4 - x)(x – 2) (4 – x)(x + 1) = (x – 2) (x + 1)
  • 11. OK? Good. Complete the class work & submit tomorrow. Last Practice; Simplify 2 2 4 4 4 x x x          2 2 2 2 x x x x     x 2 2 x    