SOLUTION: Use interval notation to express the solution set of the inequality. |2x − 5| ≥ 1

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 [,).

    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.