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The given absolute value inequality
|2x-5| >= 1 (1)
is equivalent to the system of two linear inequalities
2x - 5 >= 1 (2)
OR
-(2x-5) >= 1. (3)
Notice that the two inequalities (2) and (3) are connected by the service word " OR ", which means that the final set of solutions
is the UNION of the solution sets for each separate inequalities (2) and (3).
So, I will solve each inequality (2) and (3) separately.
(a) 2x - 5 >= 1 ====> 2x >= 1 + 5 ====> 2x >= 6 ====> x >= 6/2 = 3.
(b) -(2x-5) >= 1 ====> -2x + 5 >= 1 ====> 5 - 1 >= 2x ====> 2x <= 4 ====> x <= 4/2 = 2.
Thus the solution of the given inequality (1) is the union of two semi-infinite segments { x >= 3} and { x <= 2 }, or in interval notation
the solution set is (,] U [,).
Solved.
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The major lesson to learn from the solution is THIS :
The given absolute value inequality (1) is equivalent to the system of two linear inequalities (2) and (3)
connected by the service word " OR ", meaning the union of solution sets for linear inequalities.