Rational inequalities

**xsavethesporksx** · February 25th 2010, 11:20 PM

3 different questions:

${\left({x}+{105}\right)}{\left({x}-{178}\right)}{\left({{x}}^{{2}}-{16}\right)}\gt{0}$

${\left({{x}}^{{2}}-{3}{x}\right)}{\left({x}+{7}\right)}\le-{1}{{x}}^{{2}}+{3}{x}$

$\frac{{{2}{{x}}^{{2}}-{29}}}{{{x}-{3}}}\ge{x}-{1}$

Put into interval notation.

question on the 1st one.. Should i factor out (X-4)(X+4) or leave it as is?
third one... do i subtract (x-1) from both sides... multiply by the LCD (x-3) ... that gives me (3x^2-4x-26)/(x-3) then using quadradic i get (2+-sqrt(82))/3..

Help please

**Sudharaka** · February 25th 2010, 11:47 PM

Originally Posted by xsavethesporksx

3 different questions:

${\left({x}+{105}\right)}{\left({x}-{178}\right)}{\left({{x}}^{{2}}-{16}\right)}\gt{0}$

${\left({{x}}^{{2}}-{3}{x}\right)}{\left({x}+{7}\right)}\le-{1}{{x}}^{{2}}+{3}{x}$

$\frac{{{2}{{x}}^{{2}}-{29}}}{{{x}-{3}}}\ge{x}-{1}$

Put into interval notation.

question on the 1st one.. Should i factor out (X-4)(X+4) or leave it as is?
third one... do i subtract (x-1) from both sides... multiply by the LCD (x-3) ... that gives me (3x^2-4x-26)/(x-3) then using quadradic i get (2+-sqrt(82))/3..

Help please

Dear xsavethesporksx,

First step is to take all the terms into one side and factor the expresson. Then you could see for which values the inequality holds.

For example,

$(x+105)(x-178)(x^{2}-16)>{0}$

$(x+105)(x-178)(x-4)(x+4)>{0}$

$[x-(-105)](x-178)(x-4)[x-(-4)]>{0}$

Now draw a number line, you could consider the following situations one by one.

When, $x<-105\Rightarrow{(-)(-)(-)(-)}$ Therefore the result will be positive.

When, $-105<x<-4\Rightarrow{(+)(-)(-)(-)}$ Therefore the result is negative.

When, $-4<x<4\Rightarrow{(+)(-)(-)(+)}$ Therefore the result is positive.

When, $4<x<178\Rightarrow{(+)(-)(+)(+)}$ Therefore the result is negative.

When, $x>178\Rightarrow{(+)(+)(+)(+)}$ Therefore the result is positive.

Therefore $-105>x~or~-4<x<4~or~x>178$

$x\in{(-\infty,-105)\cup{(-4,4)}\cup{(178,\infty)}}$

Hope this will help you. Similarly you could solve your other inequalities. But if you have any questions regarding this matter please don't hesitate to reply me.

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