ALGEBRA - solved problems
  Linear inequalities
     Solving linear inequalities
86.
   Solve given linear inequality.
Solution:
                     x - 8 < 2x - 4
                      - x < 4 | (-1)
                          x > -4
 
The solution of the inequality is every real number greater than  -4.
87.    Solve inequality   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.
88.    Solve given linear inequality.
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.
89.     Solve inequality   (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 (double) inequalities
90.    Solve inequality    -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
 
91.
   Solve given linear inequality.
Solution:
Systems of linear inequalities
The solution of the system of simultaneous inequalities is the intersection of sets of the individual solutions.
92.    Solve and graph the solution of the given system of simultaneous inequalities.
Solution:
 
 
 
 
 
 
   
the solution of the system is the open interval (3, + oo ).
Solving and graphing systems of linear inequalities in two variables
The set of points whose coordinates (x, y) satisfy the inequality  ax+ by + c > 0 is a half-plane of a Cartesian plane.
93.    Solve and graph the solution of the given system of linear inequalities in two variables.
Solution:  
Plug the coordinates of the origin
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