Rational equation & inequality

1. MATHEMATICAL KEY POINTS…
 A rational expression is an expression that can be written as a ratio of two polynomials.
EXAMPLE:
(1)
(2)
 A rational equation contains rational expressions.
EXAMPLE:
(1)
(2)
 A rational inequality contains rational expressions is referred to as rational inequality.
EXAMPLE:
(1)
(2)
Procedures on Solving Rational
Equations
Procedures on Solving Rational
Inequalities
(1) Eliminate the denominators by multiplying
each terms of the equation by the least
common denominator.
(2) Note that eliminating denominators may
introduce extraneous solutions. Check the
solutions of the transferred equations with the
original equation.
(1) Rewrite the inequality as a single fraction on
one side of the inequality symbol and zero on
the other side.
(2) Determine over what the intervals of the
fraction takes on positive and negative values
EXAMPLE:
Solve for x:
EXAMPLE:
Solve for x:

2. SOLUTION:
*the LCD of all the denominators is .
( ) ( ) ( )
SOLUTION:
*the LCD of all the denominators is .
( ) ( )
NOTE:
(A) Multiplying both sides of an inequality by a
positive number it retains the direction of the
inequality
(B) Multiplying both sides of an inequality by a
negative number it reverses the direction of
the inequality