document.write( "Question 825159: Find the set of values of x for which the following inequality is true 2x-3/x-5<3/2, x can not equal 5. Please help.. \n" ); document.write( "
Algebra.Com's Answer #497129 by math-vortex(648)\"\" \"About 
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document.write( "Hi, there--\r\n" );
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document.write( "Problem:\r\n" );
document.write( "Find the set of values of x for which the following inequality is true \"%282x-3%29%2F%28x-5%29%3C3%2F2\",  with \"x%3C%3E5\".\r\n" );
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document.write( "Solution:\r\n" );
document.write( "We have the inequality, \"%282x-3%29%2F%28x-5%29%3C3%2F2\",  with \"x%3C%3E5\". We have the stipulation that \r\n" );
document.write( "\"x%3C%3E5\" to avoid division by zero.\r\n" );
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document.write( "Let's work with the related equality, \"%282x-3%29%2F%28x-5%29=3%2F2\". Then we'll use algebra to simplify \r\n" );
document.write( "it. Multiply both sides by x-5.\r\n" );
document.write( "\"%28x-5%29%2A%28%282x-3%29%2F%28x-5%29%29=%283%2F2%29%2A%28x-5%29\"\r\n" );
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document.write( "On the left hand side the factor (x-5) in the numerator and the denominator cancel out \r\n" );
document.write( "because \"%28x-5%29%2F%28x-5%29=1\". Now we have\r\n" );
document.write( "\"2x-3=%283%2F2%29%2A%28x-5%29\"\r\n" );
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document.write( "Apply the distributive property to remove the parentheses on the right hand side.\r\n" );
document.write( "\"2x-3=%283x-15%29%2F2\"\r\n" );
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document.write( "Multiply both sides by 2 to clear the fraction on the right side.\r\n" );
document.write( "\"2%2A%282x-3%29=2%2A%28%283x-15%29%2F2%29\"\r\n" );
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document.write( "The factor 2 in the numerator and denominator on the right side cancel out because \r\n" );
document.write( "\"2%2F2=1\". Apply the distributive property to the left side to clear the parentheses. Now we \r\n" );
document.write( "have\r\n" );
document.write( "\"4x-6=3x-15\"\r\n" );
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document.write( "Subtract 3x from both sides. Simplify (4x-3x=x and 3x-3x=0).\r\n" );
document.write( "\"4x-6-3x=3x-15-3x\"\r\n" );
document.write( "\"x-6=-15\"\r\n" );
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document.write( "Add 6 to both sides of the equation to isolate x on the left. Simplify (-6+6=0 and -15+6=-9).\r\n" );
document.write( "\"x-6%2B6=-15%2B6\"\r\n" );
document.write( "\"x=-9\"\r\n" );
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document.write( "The equation x=-9 is equivalent to the original equation, but much easier to understand. It \r\n" );
document.write( "represents the critical number where the original inequality shifts from true to false. \r\n" );
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document.write( "Now we will test the INEQUALITY with a sample number from either side of -9, one greater \r\n" );
document.write( "than -9, the other less than -9. We could choose any numbers; we try to choose easy ones. \r\n" );
document.write( "Let's start with 0 which is greater than -9.\r\n" );
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document.write( "Let x=0 \r\n" );
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document.write( "\"%282x-3%29%2F%28x-5%29%3C3%2F2\"\r\n" );
document.write( "\"%282%280%29-3%29%2F%28%280%29-5%29%3C3%2F2\"\r\n" );
document.write( "\"%28-3%29%2F%28-5%29%3C3%2F2\"\r\n" );
document.write( "\"3%2F5%3C3%2F2\"\r\n" );
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document.write( "x = 0 makes the inequality TRUE. This tells us that all values of x such that x is greater the -9 make the inequality true.\r\n" );
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document.write( "Now we test a value less than -9, say x=-10\r\n" );
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document.write( "Let x=0 \r\n" );
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document.write( "\"%282x-3%29%2F%28x-5%29%3C3%2F2\"\r\n" );
document.write( "\"%282%28-10%29-3%29%2F%28%28-10%29-5%29%3C3%2F2\"\r\n" );
document.write( "\"%28-23%29%2F%28-15%29%3C3%2F2\"\r\n" );
document.write( "\"23%2F15%3C3%2F2\"\r\n" );
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document.write( "We need common denominators to evaluate this inequality. Make the denominators be 30.\r\n" );
document.write( "\"46%2F30%3C45%2F30\"\r\n" );
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document.write( "We see that this inequality is FALSE. This tells us that all values of x such that x<-9 make the inequality false.\r\n" );
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document.write( "What about -9 itself? Let's try it.\r\n" );
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document.write( "Let x=-9 \r\n" );
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document.write( "\"%282x-3%29%2F%28x-5%29%3C3%2F2\"\r\n" );
document.write( "\"%282%28-9%29-3%29%2F%28%28-9%29-5%29%3C3%2F2\"\r\n" );
document.write( "\"%28-21%29%2F%28-14%29%3C3%2F2\"\r\n" );
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document.write( "Reduce the fraction on the left to lowest terms.\r\n" );
document.write( "\"3%2F2%3C3%2F2\"\r\n" );
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document.write( "When x=-9 both sides of the inequality are equal; -9 is not a solution to the inequality because 3/2 is not less than 3/2.\r\n" );
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document.write( "That's it. The set of values that makes your inequality true is all values of x such that x<-9.\r\n" );
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document.write( "Hope this helps!\r\n" );
document.write( "Mrs. Figgy\r\n" );
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