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You can put this solution on YOUR website!
you start with (x-3)/(x+4) >= (x+2)/(x-5)
\n" ); document.write( "subtract (x+2)/(x-5) from both sides of the equation to get:
\n" ); document.write( "(x-3)/(x+4) - (x+2)/(x-5) >= 0
\n" ); document.write( "x cannot be equal to -4 nor can it be equal to 5 because when it is those values, the equation becomes undefined since you can't divide by 0.
\n" ); document.write( "it appears that this equation will have asymptotes at those values of x.
\n" ); document.write( "here's a graph of the equation.
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\n" ); document.write( "it appears that the graph is greater than or equal to 0 when:
\n" ); document.write( "x < -4
\n" ); document.write( "1/2 <= x < 5
\n" ); document.write( "this means that (x-3)/(x+4) >= (x+2)/(x-5) when x < -4 and when 1/2 <= x < 5.
\n" ); document.write( "it can be seen easily by graphing the equation.
\n" ); document.write( "it's not so simple to solve algebraically as i found out much to my dismay.
\n" ); document.write( "in fact, if i didn't graph it, i would probably not have been able to solve it unless i knew how to solve problems like this from previous experience.
\n" ); document.write( "now that i did graph it, however, i could see what was happening and then able to provide an algebraic explanation for the answer.
\n" ); document.write( "i examined the original equation and determined that the equation was undefined at x = -4 and x = 5.
\n" ); document.write( "i would have needed to test if there were asymptotes at those locations.
\n" ); document.write( "that test would have required substituting values of -4.1 and -3.9 and 4.9 and 5.1 for x to see what the values of y were at those points. that would have shown that the asymptote existed at x = -4 and x = 5.
\n" ); document.write( "i would then have had to test for a zero point of the equation by setting the equation equal to 0.
\n" ); document.write( "that would have yielded a zero point at x = 1/2
\n" ); document.write( "i would then have tested the equation when x < -4 to see that it was positive.
\n" ); document.write( "that would have led to the conclusion that (x-3)/(x+4) >= (x+2)/(x-5) was true when x < -4.
\n" ); document.write( "i would then have tested the equation when x > -4 and < 1/2.
\n" ); document.write( "that would have led to the conclusion that (x-3)/(x+4) >= (x+2)/(x-5) was false during that interval.
\n" ); document.write( "i would then have tested the equation when x >= 1/2 and < 5. that would have led to the conclusion that (x-3)/(x+4) >= (x+2)/(x-5) was true when x >= 1/2 and < 5
\n" ); document.write( "i would then have tested the equation when x > 5. that would have led to the conclusion that (x-3)/(x+4) >= (x+2)/(x-5) was false when x > 5.
\n" ); document.write( "it's a lot of work.
\n" ); document.write( "graphing the equation was so much easier and allowed me to see what was happening right away.
\n" ); document.write( "the only thing i needed to know was how to graph the equation.
\n" ); document.write( "the original equation was:
\n" ); document.write( "(x-3)/(x+4) >= (x+2)/(x-5)
\n" ); document.write( "i subtracted (x+2)/(x-5) from both sides of the equation to get:
\n" ); document.write( "(x-3)/(x+4) - (x+2)/(x-5) >= 0
\n" ); document.write( "i then set the expression on the left side of that equation equal to y to get:
\n" ); document.write( "y = (x-3)/(x+4) - (x+2)/(x-5)
\n" ); document.write( "this is the equation that i graphed.
\n" ); document.write( "the rest was just looking and seeing where the equation was positive and where it was not, keeping in mind the asymptotes at x = -4 and x = 5.
\n" ); document.write( "the basic logic used was:
\n" ); document.write( "if a >= b, then (a - b) >= 0\r
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