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Transcript
THE RATIONAL NUMBERS
2-1 Rational Numbers
A rational number is a terminating or repeating decimal.
All rational numbers can be written as fractions.
Multiplicative inverse:
Converting a repeating decimal to a fraction:
2-2 Simplifying Rational Expressions
A rational expression is a fraction where the numerator and denominator are polynomials.
x 1
2
x2  4 x  4
3x
Ex: 2
x
x 1
2x
y
A rational expression is undefined when the denominator equals zero.
2
2
2

Ex:
is undefined when x  1 because
x 1
1  1 0
Like any fraction a rational expression can be simplified by canceling factors common to the
numerator and the denominator.
12 3  4 3
Ex:


8 2 4 2
12 x 2 y 3  4  x  x  y 3x


8 xy 2
2 4  x  y  y 2y
x2
x2
1


x  4x  4  x  2  x  2 x  2
2
** Only terms that are being MULTIPLIED can be canceled. You CANNOT cancel terms that are
being added or subtracted.
x
x

x2 x 2
x
cannot be simplified.
x2
** Always remember to factor completely before you cancel
2-3 Multiplying and Dividing Rational Expressions
To multiply rational expressions, multiply the numerators and multiply the denominators.
 x  2   y  y  x  2  xy  2 y

Ex: 


 x   y  3  x  y  3 xy  3x
Answers must be shown in simplest form. You can simplify after you multiply, but it is easier to
cross-cancel before you multiply.
 3x  6   y 2  3 y   3  x  2    y  y  3   x  2  y  3 xy  2 y  3x  6
Ex: 





2x
2x
3 x 
 2 y  3x   2 y  
To divide rational expressions, multiply the fist term by the reciprocal of the second term.
 x  3   x  3   x  3  4  2  2 2
Ex: 



 

 2x   4   2x   x  3  2 x x
2-4 Adding and Subtracting Rational Expressions
To add and subtract rational expressions you must first find common denominators.
 x  2  x  2   x  x  2 
x2
x


Ex:
x  2 x  2  x  2  x  2   x  2  x  2 


x2  4 x  4 x2  2 x
 2
x2  4
x 4
2
 x  4x  4   x2  2x 
x2  4
2 x2  2 x  4

x2  4
2-5 Ratio and Proportion
A ratio is a comparison of values. A proportion is an equation of two equal ratios.
Cross-multiply to solve a proportion.
x
x 1

Ex:
x2
x
x  x    x  1 x  2
x 2  x 2  3x  2
0  3x  2
3 x  2
2
x
3
2-6 Complex Rational Expressions
Complex fractions are fractions within fractions.
Ex:
1
x
x 1
2x
3
x1
2
3
7
x
To simplify a complex fraction re-write the fraction as a division problem. Then divide and simplify.
1
1
1
 1  1 
x
   x  1   
Ex:
 2
x 1 x
 x  x  1  x  x
Ex:
2x
3
x 1
2

2 x x  1  2 x  2 
4x

  

3
2
 3  x  1  3x  3
2-7 Solving Rational Equations
There are several ways to solve rational equations. You can:
1. Work with fractions.
2. Multiply the equations by the LCD to remove fractions.
3. Combine fractions on each side of the equation to create proportion.
**Any solution that causes fractions in the original equation to be undefined is called an extraneous
solution. Extraneous solutions must be rejected.
2-8 Solving Rational Inequalities
Rational inequalities can be solved using the same methods as rational equations. Remember
that when you multiply or divide by a negative number the inequality symbol changes direction.