Linear Inequalities
     
     Linear inequalities
      Inequality signs
      Properties of inequalities
      Solving linear inequalities
Inequality signs
Inequality is a statement about the relationship between two numbers or quantities that holds when they are comparable and related by ordering but are not equal.
These relationships are written as:
                 a < b  meaning a is less than b,
                 a meaning a is less than or equal to b,
                 a > b  meaning a is greater than b,
                 a meaning a is greater than or equal to b.
Properties of inequalities
The trichotomy property of the real line states that for any real numbers, a and b, exactly one of
a < b a = ba > b,  is true.
The transitive property of inequalities states that for any real numbers, a, b and c,
             if  a < and  b < c,  then  a < c,
             if  a > and  b > c,  then  a > c.
The addition and subtraction property of inequalities states that for any real numbers, a, b and c,
             if  a < b,  then  a + c < b + c,  and  a - c < b - c,
             if  a > b,  then  a + c > b + c,  and  a - c > b - c.
The multiplication and division property of inequalities states that for any real numbers, a, b and c,
             if c is positive and  a < b, then  a c < b c,  and   a/c < b/c,
             if c is negative and  a < b, then  a c > b c,  and   a/c > b/c.
The sense of an inequality is not changed if both sides are increased or decreased by the same number, or if both sides are multiplied or divided by a positive number.
The sense of an inequality is reversed if both sides are multiplied or divided by a negative number.
The additive inverse property of inequalities states that for any real numbers a and b,
             if  a < b then  -a > -b,
             if  a > b then  -a < -b.
The multiplicative inverse property of inequalities states that for any real numbers a and b that are both positive or both negative,
             if  a < b then  1/a > 1/b,
             if  a > b then  1/a < 1/b.
Note that, the above properties also hold if strict inequality signs, < and >, are replaced with their corresponding non strict inequality signs, and .
Solving linear inequalities
A linear inequality has the standard form   ax + b > 0,  where a, b R
and where other inequality signs like  <, > and < can appear.
As with equations, the inequality is solved when positive x is isolated on the left.
The solutions to an inequality are all values of x that make the inequality true.
We may add the same number to both sides of an inequality.
By multiplying both sides of an inequality by the same negative number, the sense of the inequality changes, 
i.e., it reverses the direction of the inequality sign.
Example:
                     x - 8 < 2x - 4
                      - x < 4 | (-1)
                          x > -4
 
The solution of the inequality is every real number greater than  -4.
Beginning Algebra Contents C
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