Absolute value equations
         Solving absolute value equations
      Linear Inequalities
         Solving inequalities
         Properties of inequalities
         Examples of solving single linear inequalities
         Solving compound (double) inequalities
Absolute value equations
Solving absolute value equations
Recall that the absolute value of a real number a, denoted | a |, is the number without its sign and represents the distance between 0 (the origin) and that number on the real number line.
Thus, regardless of the value of a number a its absolute value is always either positive or zero, never negative that is,  | a | > 0.
To solve an absolute value equation, isolate the absolute value on one side of the equation, and use the definition of absolute value.
If the number on the other side of the equal sign is positive, we will need to set up two equations to get rid of the absolute value,
  - the first equation that set the expression inside the absolute value symbol equal to the other side of the equation,
  - and the second equation that set the expression inside the absolute value equal to the opposite of the number on the other side of the equation.
Solve the two equations and verify solutions by plugging the solutions into the original equation.
If the number on the other side of the absolute value equation is negative then the equation has no solution.
Example:     | 1 - 2x | = 17
Solution:        1 - 2x = 17                             or                      1 - 2 x = -17
                           2 x = - 16                                                         2x = 18
                             x = - 8                                                             x = 9
The solutions to the given equation are x = - 8 and x = 9.
Example:     | -3 - x | = 5
Solution:         -3 - x = 5                             or                      -3 - x = -
                              x = - 8                                                            x = 2
The solutions to the given equation are x = - 8 and x = 2.
Example:     | x + 1 | = 2x - 3
Solution:        x + 1 = 2x - 3                             or                      x + 1 = -(2 x - 3)
                     x - 2x = - 4                                                         x + 2x = 3 - 1
                             x = 4                                                                   3x = 2     =>     x = 2/3
Check solutions:
   x = 4   =>     | x + 1 | = 2x - 3,                                x = 2/3   =>      | x + 1 | = 2x - 3 
                        | 4 + 1 | = 2 4 - 3                                                   | 2/3 + 1 | = 2 2/3 - 3
                                  5 = 5                                                                    5/3  is not equal  -5/3
The check shows that x = 2/3 is not a solution, because the right side of the equation becomes negative. There is a single solution to this equation: x = 4.
Example:     | x + 2 | = | 2x - 5 |
Solution:   As both sides of the equation contain absolute values the only way the two sides are equal is, the two quantities inside the absolute value bars are equal or equal but with opposite signs.
                      x + 2 = 2x - 5                             or                      x + 2 = -(2 x - 5)
                     x - 2x = - 5 - 2                                                     x + 2 = -2 x + 5
                            -x = -7                                                                 3x = 3     
                              x = 7                                                                    x = 1
Check solutions:
   x = 7   =>     | x + 2 | = | 2x - 5 |,                           x = 1    =>     | x + 2 | = | 2x - 5 | 
                        | 7 + 2 | = | 2 7 - 5 |                                               | 1 + 2 | = | 2 1 - 5 |
                                  9 = 9                                                                        3 = | -3 |
Therefore, the solutions to the given equation are  x = 7 and  x = 1.
Linear Inequalities
A linear inequality is one that can be reduced to the standard form ax + b > 0 where a, b R, and where other inequality signs like < ,  >  and  <  can appear.
Solving inequalities
The solutions to an inequality are all values of x that make the inequality true. Usually the answer is a range of values of x that we plot on a number line.
We use similar method to solve linear inequalities as for solving linear equations:
 - simplify both sides,
 - bring all the terms with the variable on one side and the constants on the other side,
 - and then multiply/divide both sides by the coefficient of the variable to get the solution while applying following properties:
Properties of inequalities
1. Adding or subtracting the same quantity from both sides of an inequality will not change the direction of the inequality sign.
2. Multiplying or dividing both sides of an inequality by a positive number leaves the inequality symbol unchanged.
3. Multiplying or dividing 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.
Examples of solving single linear inequalities
Solve each of the following inequalities, sketch the solution on the real number line and express the solution in interval notation.
Example:     3(x - 2) > -2(1- x)
Solution:     3x - 6 > -2 + 2x
                           x > 4
 
       interval notation (4, oo)
The open interval  (4, oo) contains all real numbers between given endpoints, where round parentheses indicate exclusion of endpoints.
Example:   
Solution:        -4x + 9 - 3x < 6 - 5 + 5x 
                                   -12x < -8
                                         x > 2/3
 
interval notation
The half-closed (or half-open) interval contains all real numbers between given endpoints, where the square bracket indicates inclusion of the endpoint 2/3 and round parenthesis indicates exclusion of infinity.

Example:    (x - 3) (x + 2) > 0

Solution:   The factor x - 3 has the zero at x = 3, is negative for x < 3 and is positive for x > 3, and
                 the factor x + 2 has the zero at x = -2, is negative for x < - 2 and is positive for x > -2,
as is shown in the table
x - oo increases -2 increases 3 increases + oo
x + 2   - 0 + + +  
x - 3    - - - 0 +  
(x - 3)(x + 2)   + 0 - 0 +  
Thus, the given inequality is satisfied for    - oo < x < - 2    or     3 < x < + oo
                     in the interval notation     ( - oo , -2 ] U [ 3, + oo )
Solving compound or double inequalities
Use the same procedure to solve a compound inequality as for solving single inequalities.
Example:      -4 < 2(x - 3) < 5
Solution:  We want the x alone as middle term and only constants in the two outer terms. Remember, while simplifying given compound inequality, the operations that we apply to a middle term we should also do to the both left and right side of the inequality.
                      -4 < 2(x - 3) < 5 | 2
 
Example:          
Solution:
 
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