Solving Inequalities by f8cM01

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									Solving Inequalities
        Solving Inequalities
• Solve Inequalities ≡ find value(s) of the
  variable(s) that make the inequality true
• Inequalities almost always generate an
  infinite number of solutions (a set or
  interval of numbers on the R Number line)
• Testing answer(s) is more complex:
    • You can’t test every one
    • Test boundaries w/ corresponding equation
    • Test points in each interval (sign graph)
        3 Ways to Represent Solutions
                                                                 Compound:
                                                               or F either, union
1. Inequality notation uses <, , >, :                     and F both, intersection

     x4          x0           x 1   or   x4      3  x  7
                                                   x  3 and x  7
                    unbounded                          bounded

•   Interval notation uses (a,b), [a,b], (a,b], [a,b):
    – a and b are the endpoints of the interval and a  b
    – If an endpoint is included, a square bracket is used: [ or ]
                      is not included, a parenthesis is used: ( or )
                      is ±∞, it is never included (±∞  Real numbers)

    [4, )      (,0)          (,1)  [4, )          (3,7]
                                                     (,7]  (3, )
                   unbounded                             bounded
3. Graph on the number line:
    ex: x  4 or [4, ∞)
                                 0           4
 Try these: Write in inequality and interval notation
            and graph on number line.
1.   x is greater than -5.



2.   y is at most 24.



3.   x is greater than 2 and at most 6



4.   x is less than or equal to -1 or greater than 16
                 Weird or Boundary cases
• Open-Open Bounded Interval Notation can be mistaken for a point:
   Is (3,4) an ordered pair representing (x,y) or f (3)  4 [function notation] ?
         or
    Is (3,4) the open interval from 3 to 4, i.e., 3 < x < 4?

                         Context is important



• Compound inequalities (“or” or “and”) can lead to solutions that are:
    all Reals:                      or the empty set:
                                                               Notice the violation of
            x  0 or x  3                    4 x2          the Transitive Property!
                x                        x  4 and x  2
                                                 
         Strategies to Solve Inequalities
Type             Strategy
Linear           Isolate the variable by simplifying and using the Addition and
                 Multiplication Properties of Inequality. (~linear equalities)
                 *** Reverse the direction of the inequality symbol if you multiply or
                 divide by a negative quantity. ***
Absolute Value   Isolate the absolute value and then use the definition of absolute value
                 to create two cases. (~absolute value equalities).
                 If ,  then join by “or”. If ,  then join by “and”.
Other types:     Find critical numbers where the expression changes sign or is
                 undefined.
                      Sign changes are found by converting to standard form and solving the
                      corresponding equality (= 0).
                 The critical numbers create intervals on the number line to be tested in
                 the original inequality.
 Polynomial     Find critical numbers (solve corresponding equation).
 (quadratic or  Test values w/in each interval to determine the intervals that make the
 higher degree) original inequality true (sign graph).
 Rational        Place in standard form and combine fractions into one rational
                 expression (do not clear fractions as you would for Equalities).
                 Find critical numbers: where numerator is 0 (sign change) and where
                 denominator is 0 (undefined).
                 Test values w/in each interval to determine the intervals that make the
                 original inequality true (sign graph).

								
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