How Do We Add And Subtract Rational Expressions ?

To add and subtract rational expression, you have to first find a common denominator.

                                                    X-1                X-5        <------ Numerator
                                                  X +6                X+6       <------ Denominator
Lets take :

              u-v               6u -3v
              8v       +          8v 
 The first step in solving this is finding a common denominator. Since this rational expression already is in a common denominator, you don't have to worry about it.

Next you add or subtract the terms, according to what the question ask you. In this case, You add by adding like terms.

          So .....                  u + 6u = 7u   and  -1v-3v = -4v 
                Together the numerator is  7u-4v
                               the denominator is 8v.


NOW TRY THIS : 

                     7p+9z     -     3p+8z
                         9g                 9g

How Do We Solve Fractional Exponent Equations ?

To solve fractional rational exponents you have to get the variable and exponent alone before first.

 Let's use :  x^4/3 - 6 = 10 .

The first step is to add 6 to both sides to get x^4/3 by itself .
        The equation now looks like : x^4/3 = 16
The next step is to find the reciprocal of the exponent ( 4/3 ) and multiply it to both sides. This gets x by its self.

       Visual :  (x^4/3) (3/4) = 16 (3/4)
                          x = 16 ^ (3/4)
Next you have break up the exponent. You do this by putting 16 ^ (1/4) first. Then squaring that by 3.

           Visual :                               16^ (1/4) = 2.
                                            2^3 = 8 
                                 Your final answer is 8 . 

Now Try It Yourself :
                                        8^(2/3)

* Remember : (1/2) = square roots 
                        (1/3) - cubed root 
                                           
                   

How Do We Solve Radical Equations ?

Lets use the example :
The first step in solving radical equations is making sure the radical is by itself.

Let's take √x-7 + 5 = 6 , you would have to subtract 5 from each side to get the radical by itself. 
The next step is to square each side of the equation. Do not square terms. 

   Here is an example : 

               CORECT : √x-7^2 =1^2
        
                              x-7 =1
The next step is to solve the equation by solving for X. You can do that by getting X alone. In this case it would look like : 

                                     x-7 =  1
                           +7    +7
                              x = 8
The last and final step is to plug the value of X back into the equation to see if it works.


                          √ 8-7 + 5 = 6 ----------> 8-7 is 1

                   √1+ 5  = 6  -----> The square root of 1 is 1.

                           6 = 6  ----------> 1+5 =6

                    The solution X = 8 is true . 



Now try it yourself  :


                             √x-3 = 5

How Do We Factor By Grouping ?

To factor by grouping, you have to use 5 easy and simple steps.
     Let's use the polynomial : 8x^2-10x-3 
The first step is to find the Master Product. The master product is multiplying the first number by the last number. So in this case you would multiply 8 by -3 and get -24.
The next step if to find what multiplies to give you -24 and adds together to give you -10.
                                    The two numbers become -12 and 2.
Next you replace the -10x with the two new factors you found ( -12 and 2 )
              Your polynomial should now look like this : 8x^2-12x+2x-3
 Now its time to group the the terms in two pairs. Group the first two terms and the last two terms
       You have to factor each pair by finding the greatest common factor.
           So you can pull out 4x from 8x^2 -12 . The first two terms now look like :
                                                 4x(2x-3) 
         You can pull out 1 from 2x -3 because there's nothing else that goes into both 2 and 3.
  Both terms together now looks like this :  4x(2x-3) + 1(2x-3)
      The final and last step is to factor out the shared binomial, which is (2x-3) and combine the term in front of the parenthesis.
                       Your final answer is (4x+1) (2x-3) .

Now try it is your self  :

                                              6x^2 + 19x + 10

" How Do We Calculate Quadratic Inequalities? "

                      Calculating quadratic inequalities are quite easier than it sounds !
                 You can solve quadratic equations in 3 short, easy, and simple steps !
 The first step in solving quadratic inequalities is taking a inequality like "  y > 2x^2 " and turning it into a equation by replacing the inequality sign with an equal sign. 
                    so " y > 2x^2 " turns into " y = 2x^2 " 

     The second step is to graph the equation " y = 2x^2 " using a dashed line for  < or > and a solid line for < or >.        

     The third step would be to choose a test point, in or outside the parabola.
               Let's use the test point ( 2, 0 ) . You plug the test point into the inequality. 
                                   y > 2x^2     
                            0 > 2(2)^2
                            0 > 2(4)
                            0 > 8   <-- This is false : 0 is not greater than 8. 
    The last step would be to shade in the side (whether the inside of the parabola or outside the parabola).

 * not accurate to problem , just visual example * 

 * where ever your test point is, if it is true you shade the whole thing . If it is false than you should the opposite * .      
            The final answer to the inequality is the shaded area 

NOW TRY IT YOURSELF : 
                      USING THE INEQUALITY : " y > x^2 - 1 " 
                                                                                                                                     

" How Do We Complete the Square ? "

Completing the square is as easy as it sounds !
                 Let's consider the equation: "  x^2+6x+2=0 "
    The first step in completing the square is is moving the constant term, which in this case is 2, to the other side of the equation. The equation now looks like: "  x^2+6x+__= -2 "
     The second step is to take the coefficient of the middle term, which is 6 in this case, and dividing it in half and squaring it. So 6 divided in half is 3. 3 squared is 9.
     The third step is to add 9 to both sides of the equation. " x^2+6x+9 = -2+9 "
                                                                 simplified to :      x^2 + 6x + 9 = 7
  The fourth step is to factor the trinomial : (x+3)^2 = 7
 The next step is to find the square root of both sides : The square root of (x+3)^2 is (x+3) .

      The square root of 7 can be either negative or positive , so its written like: +-√7
                                                                                


                                         
    The last step is to subtract 3 from each side : (x+3) =  +- √7
                                         - 3         - 3





      The final answer you get is : x= -3 +- √7




Now You Try It :
                                  x^2 + 8x + ___ = 7 
                                                                   
                                                             

How Do We Use Imaginary Numbers ?

             Using imaginary numbers is quite simple. First you need to know that " i " is defined as  " √-1 "
                                                         " i=√-1 "
       This is also known as the imaginary unit. We use imaginary numbers, i , to solve the square

 root of negative numbers. For example, we have the square root of -9. The first step to solve 

this is to make -9 positive by making -9 equal to the square root of -1 and positive 9. Knowing 

that the square root of -1 is equal to i, you replace √-1 wit i . Knowing that 9 is a perfect square, 

you can simplify it to 3. When working with imaginary numbers, i come after the real number 

only under the exception that the number is still in square root form. The final answer for the √-9 

is 3i. 


              

         Here's a visual of what i just explained : 

                                     
                                √-9 = √-1 √9

                                      = i√9
                                                            
                                      = 3i 



Now Try It Yourself Using: 


                                √-25

" Why do we flip the inequality symbol when multiplying by a negative number or solving absolute value inequalities?"

         In class, we are always told to flip the inequality symbol when multiplying by a negative number or solving an absolute value inequality. Many of us don't know why we have to flip the inequality symbol. The reason you have to flip the inequality symbol is because when you multiply by a negative number it changes the  way the problem is read.

For instance, take " -5<2 " , which reads -5 is less than 2. This is true because when you look at a number line -5 is less than 2. If I was to multiply both sides by -2, -5 becomes 10 and 2 becomes -4. Writing 10<-4 or 10 is less than -4, is inaccurate. You have to flip the sign to make the inequality true.

          Another example of why you have to flip the inequality sign is if you have 6<12 or 6 is less than 12,and multiply both sides by -3, 6 becomes -18 and 12 becomes -36. Writing -18<-36 or -18 is less than -36 is inaccurate. You have to flip the sign to read -18>-36 or -18 is greater than -36, because on a number line -36 is further from 0 than -18 is, which makes -36 smaller than -18.

Let's check your understanding :

                                                                     Using: 2>-3
                                        Multiply both sides by -3 . What is the new inequality ?