## Solving Rational Inequalities: 1/4(x+2) <= -3/4(x-2)

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DG1
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### Solving Rational Inequalities: 1/4(x+2) <= -3/4(x-2)

Related to this 2011 thread.]

I have the following rational inequality

1/4(x+2) <= -3/4(x-2)
By adding +3/4(x-2)
I simplify to (x+1)/(x^2 - 4)
This gives intervals for f(x)<=0 as (-ve infinity to -2) and [-1 to 2)
+2 & -2 being excluded as they are the vertical asymptote so

Question

a) if I solve the same inequality by cross multiplying the denominator of term on the left to the numerator of the term on the right and cross multiplying denominator of term on the right to the numerator of the term on the left and remove the 4 as they are on both sides, I get

((x-2)/(x+2)) + 3 <=0
(x-2+3x+6)/(x+2) <=0
(4x+4)/(x+2) <=0
Which I can simplify to
2(x+2)/(x+2)<=0
2<=0

This is not true. So why is the 2nd approach giving me this incorrect result? If this is not permissible, pl can you explain why?

Thank you
Last edited by PM_Admin on Sun Oct 11, 2015 5:53 pm, edited 2 times in total.
Reason: Splitting new post from very old thread; creating useful subject line.

nona.m.nona
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### Re: Solving Rational Inequalities: 1/4(x+2) <= -3/4(x-2)

1/4(x+2) <= -3/4(x-2)
By adding +3/4(x-2)
I simplify to (x+1)/(x^2 - 4)
By what steps did you obtain this result? (Your formatting is ambiguous. Are the variable factors in the numerators or in the denominators?)
a) if I solve the same inequality by cross multiplying....
By what logic have you "cross-multiplied" in an inequality, given that you do not (yet) know the signs of the factors?

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