Saturday, March 12, 2011

Equations and Inequalities

Linear Equations
Linear equations are just equations with a degree of one (highest power of a variable is one). They can be solved simply by rearranging the terms.
Eg. 7 + 5x = -8
      5x = -15
      x = -3

      3(9 + x) = 6
      9 + x = 2
      x = -7

Equations with Squares and Roots
Equations with squares or roots can be solved like a normal equation. In some case, you may have to use methods to solving quadratic equations.
  1. Completing the square
    Adding the square of half of the coefficient of x
    Eg. x^2 -4x -12 = 0
    x^2 -4x + 4 = 16
    (x - 2)^2 = 16
    x - 2 = +/-4
    x = 6 or -2
  2. Factorising
    x^2 -4x - 12 = 0
    (x-6)(x+2)=0
    x = 6 or -2
  3. Quadratic Formula
Inequalities
Inequations can be solved just like an equations, but when ever you divide or multiply by a negative number, the sign will be flipped around.
Eg. -(x+2) > -5
      x+2 < 5
      x < 3

Simultaneous Equations
There are two methods to solve simultaneous equations but I will recommend you use the substitution method.
Eg. x^2 + 2x = 5x - 2
      x^2 - 3x + 2 = 0
      (x-2)(x-1) = 0
      x = 1 or 2

Inequalities with an unknown denominator
A bit more advanced stuff. If you had a unknown in the denominator you would have to find critical points in the inequation. A critical point is a point where the gradient of the tangent is zero or cannot be defined. The denominator of a fraction cannot be zero in this case when the denominator is equal to zero gives the critical point. The inequation can then be solved afterwards and then substitution of points is necessary. (I think I confused you)
Eg.





Find the Critical Points



Solve for x





Substitute values in to determine whether it fits in.












Hope this has helped you.

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