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Solving Rational Inequalities on a Graphing Calculator

Key Questions

How do you solve rational inequalities?

Let us solve the following rational inequality.

$f(x)={x+1}/{x^2+x-6} le 0$

Set the numerator equal to zero, and solve for $x$ .

$x+1=0 => x=-1$

(Note: $f(-1)=0$ )

Set the denominator equal to zero, and solve for $x$ .

$x^2+x-6=(x+3)(x-2)=0 => x=-3,2$

(Note: $f(-3)$ and $f(2)$ are undefined.)

Using $x=-3,-1,2$ above to split the number line into open intervals:

$(-infty,-3), (-3,-1),(-1,2)$ , and $(2,infty)$

Using sample numbers $x=-4,-2,0,3$ for each interval above, respectively, we can determine the sign of (LHS).

$f(-4)=-2<0 => f(x)<0$ on $(-infty,-3)$

$f(-2)=1/4>0 => f(x)>0$ on $(-3,-1)$

$f(0)=-1/6<0 => f(x)<0$ on $(-1,2)$

$f(3)=2/3>0 => f(x)>0$ on $(2,infty)$

Hence, $f(x) le 0$ on $(-infty,-3)cup[-1,2)$ .

(Note: $-1$ is included since $f(-1)=0$ .)

The graph of $y=f(x)$ looks like:

I hope that this was helpful.

Wataru · 1 · Oct 31 2014

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Solving Rational Inequalities

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