document.write( "Question 61887: Couls someone help please?\r
\n" ); document.write( "\n" ); document.write( "Solve the following inequalitites. Write the answers in interval notation.\r
\n" ); document.write( "\n" ); document.write( "x^2 +7x-18=>0\r
\n" ); document.write( "\n" ); document.write( "3x-4/2x+1 <=6 \n" ); document.write( "

Algebra.Com's Answer #854056 by ikleyn(53750) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Could someone help please?
\n" ); document.write( "Solve the following inequalities. Write the answers in interval notation.
\n" ); document.write( "(a) x^2 + 7x - 18 => 0
\n" ); document.write( "(b) (3x-4)/(2x+1) <= 6
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\n" ); document.write( "\n" ); document.write( " The solutions by @jai_kos are incorrect for both inequalities.\r
\n" ); document.write( "\n" ); document.write( " I came to bring correct solutions for both problems.\r
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document.write( "(a)  Your starting inequality is\r\n" );
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document.write( "         x^2 + 7x - 18 => 0.    (1)\r\n" );
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document.write( "     Factor left side quadratic polynomial\r\n" );
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document.write( "         (x+9)*(x-2) >= 0.\r\n" );
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document.write( "     Critical points are  x = -9  and  x = 2.\r\n" );
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document.write( "     On the left of the critical point  x = -9,  both factors (x+9) and (x-2) are negative, so their product is positive.\r\n" );
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document.write( "     Between the critical points -9 < x < 2, factor (x+9) is positive, while factor (x-2) is negative,\r\n" );
document.write( "             so their product is negative.\r\n" );
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document.write( "     On the right of the critical point  x = 2,  both factors (x+9) and (x-2) are positive, so their product is positive.\r\n" );
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document.write( "     Thus the solution set for inequality (1) is  x < -9  or  x > 2,  \r\n" );
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document.write( "     or, in interval notation, the union (,) U (,).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Part (a) is solved correctly.\r
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document.write( "(b)  Your starting inequality is\r\n" );
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document.write( "      <= 6.    (1)\r\n" );
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document.write( "Transform it equivalently this way\r\n" );
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document.write( "      - 6 <= 0              <<<---===  moving 6 from right side to left side with changing the sign\r\n" );
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document.write( "      -  <= 0    <<<---=== writing '6' with the common denominator\r\n" );
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document.write( "      <= 0       <<<---===  simplifying\r\n" );
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document.write( "      <= 0                <<<---===  simplifying further\r\n" );
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document.write( "Now, the left side rational function can be non-positive if and only if\r\n" );
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document.write( "    EITHER the numerator is non-negative and denominator is negative\r\n" );
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document.write( "        -9x - 10 >= 0  and  2x + 1 < 0    (2)\r\n" );
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document.write( "    OR     the numerator is non-positive and denominator is positive\r\n" );
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document.write( "        -9x - 10 <= 0  and  2x + 1 > 0.   (3)\r\n" );
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document.write( "In case (2),  9x <= -10  and  x < -1/2,  which is the same as  \r\n" );
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document.write( "              x <= -10/9 and  x < -1/2.\r\n" );
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document.write( "              These both inequalities, taken together, have the solution set  x <= -10/9.\r\n" );
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document.write( "In case (3),  9x >= -10  and  2x > -1,  which is the same as  \r\n" );
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document.write( "              x >= -10/9 and  x > -1/2.\r\n" );
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document.write( "              These both inequalities, taken together, have the solution set  x >= -1/2.\r\n" );
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document.write( "Thus the final solution to the given inequality is this set of real numbers  { x <= -10/9 } OR { x >= -1/2 }.\r\n" );
document.write( "In the interval notation, the solution set is the union of two sets (,] U (,).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Both problems/questions are solved correctly.\r
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