document.write( "Question 1103752: Solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. (If the solution set is empty, enter EMPTY or ∅.) 3y<5-2y<7+y \n" ); document.write( "

Algebra.Com's Answer #718475 by ikleyn(52799) $\"\"$ $\"About$

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document.write( "This compound inequality\r\n" );
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document.write( "3y < 5y-2 < 7+y \r\n" );
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document.write( "is EQUIVALENT to (or, if you want, is, BY THE DEFINITION)\r\n" );
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document.write( "the system of TWO INEQUALITIES\r\n" );
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document.write( "3y < 5y-2      (1)   and\r\n" );
document.write( "5y-2 < 7+y     (2)\r\n" );
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document.write( "in the sense that the final solution set is THE INTERSECTION of the solution sets for two inequalities (1) and (2).\r\n" );
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document.write( "So, let us solve the inequality (1) first.\r\n" );
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document.write( "1)  3y < 5y - 2 <=== is equivalent to (subtract 3y from both sides) ===> \r\n" );
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document.write( "    0 < 2y - 2  <=== is equivalent to (add 2 to both sides) ===> \r\n" );
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document.write( "    2 < 2y      <=== is equivalent to (divide by 2 from both sides) ===>\r\n" );
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document.write( "    y > 1.\r\n" );
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document.write( "    So, the first inequality is solved, and its solution set is y > 1,  or,\r\n" );
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document.write( "    in the interval notation, the set (1,infinity).     (*)\r\n" );
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document.write( "Next, let us solve the second inequality \r\n" );
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document.write( "2)    5 - 2y < 7 + y.  <=== It is equivalent to (add 2y to both sides) ===>\r\n" );
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document.write( "    5 < 7 + 3y      <===     is equivalent to (subtract 7 from both sides) ===>\r\n" );
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document.write( "    -2 < 3y         <===     is equivalent to (divide by 3 from both sides) ===>\r\n" );
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document.write( "     < y,          or, which is the same\r\n" );
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document.write( "    y > .\r\n" );
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document.write( "    Thus, the second inequality is solved, and its solution set is y > ,  or,\r\n" );
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document.write( "    in the interval notation, the set (,infinity).    (**)\r\n" );
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document.write( "3.  The intersection of both sets (*) and (**) is the set  y > 1,  or, in the interval notation, (1,).\r\n" );
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document.write( "Answer.  The solution of the given compound inequality is  y > 1,  or, in the interval notation, (1,).\r\n" );
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