Question 1131539
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<pre>
The given <U>absolute value</U> inequality

    |2x-5| >= 1       (1)

is equivalent to the system of two <U>linear</U> 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  ({{{-infinity}}},{{{2}}}] U [{{{3}}},{{{infinity}}}).
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Solved.


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<U>The major lesson to learn from the solution is THIS</U> :


<pre>
    The given <U>absolute value</U> inequality (1) is equivalent to the system of two <U>linear</U> inequalities (2) and (3)

    connected by the service word " OR ", meaning the union of solution sets for linear inequalities.
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