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Sunday, 24 March 2013

Intermediate Algebra chapter 2 part 4


Solving Linear Inequalities
Chapter 2,

Properties Used to Solve Inequalities
For real numbers a, b, and c:

  1. If a > b, then a + c > b + c.
  2. If a > b, then a - c > b - c.
  3. If a > b and c > 0, then ac > bc.
  4. If a > b and c > 0, thena
    c
      >  b
    c
  5. If a > b and c < 0, then ac < bc.
  6. If a > b and c < 0, thena
    c
      <  b
    c
Note that when you multiply by a negative number the inequality sign changes direction.
How to Write Solutions
 In Set Builder
Notation
On Number LineIn Interval
Notation
x > a{ x | x > a }(a, )
 a{ x | x  a }[a, )
x < a{ x | x < a }(-, a)
 a{ x | x  a }(-, a]
a < x < b{ x | a < x < b }(a, b)
 x  b{ x | a  x  b }[a, b]
a < x  b{ x | a < x  b }(a, b]
 x < b{ x | a  x < b }[a, b)
Solving Simple Inequalities
Solve a simple linear inequality like you would solve a linear equation except that the = sign is repaced by an inequality sign.
ExampleSolution Steps
Solve 5x - 2 < 35x < 5
x < 1

The solution is x < 1
Solving Compound Inequalities
A compound inequality has an expression in the middle sandwiched between two inequalities like in 5 < 2x - 3  11.
Goal: get x alone (with coefficeint of 1) in the middle of the sandwich with numbers on the outsides.
What you do to the middle (while trying to get x alone) you must do to the outside expressens as well. That will keep each step's inequality equivalent to the original inequality (equivalent means that the solutions are the same).
ExampleSolve 5 < 2x - 3 < 11

Solution Steps

In Words...
8 < 2x < 14Add 3 to each expression.
4 < x < 7Divide each expression by 2.

The solution is 4 < x < 7
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