document.write( "Question 1131539: Use interval notation to express the solution set of the inequality.
\n" ); document.write( "|2x − 5| ≥ 1 \n" ); document.write( "

Algebra.Com's Answer #748204 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "The given absolute value inequality\r\n" );
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document.write( "    |2x-5| >= 1       (1)\r\n" );
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document.write( "is equivalent to the system of two linear inequalities\r\n" );
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document.write( "    2x - 5 >= 1       (2)\r\n" );
document.write( "OR\r\n" );
document.write( "    -(2x-5) >= 1.     (3)\r\n" );
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document.write( "Notice that the two inequalities (2) and (3) are connected by the service word \" OR \", which means that the final set of solutions\r\n" );
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document.write( "is the UNION of the solution sets for each separate inequalities (2) and (3).\r\n" );
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document.write( "So, I will solve each inequality (2) and (3) separately.\r\n" );
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document.write( "    (a)   2x - 5 >= 1  ====>  2x >= 1 + 5  ====>  2x >= 6  ====>  x >= 6/2 = 3.\r\n" );
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document.write( "    (b)   -(2x-5) >= 1  ====>  -2x + 5 >= 1  ====>  5 - 1 >= 2x  ====>  2x <= 4  ====>  x <= 4/2 = 2.\r\n" );
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document.write( "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\r\n" );
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document.write( "the solution set is  (,] U [,).\r\n" );
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document.write( "    The given absolute value inequality (1) is equivalent to the system of two linear inequalities (2) and (3)\r\n" );
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document.write( "    connected by the service word \" OR \", meaning the union of solution sets for linear inequalities.\r\n" );
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