Polynomials and Rational Functions
4.5 Polynomial Inequalities
Sign-Chart Strategy
For polynomial and rational inequalities, the sign chart keeps the logic organized.
Fact: Sign-chart steps
- Move all terms to one side.
- Factor numerator and denominator when possible.
- Find critical points (zeros and undefined values).
- Test interval signs.
- Select intervals matching inequality symbol.
Example: Solve a polynomial inequality
Solve
$$x^2-5x+6>0.$$
Show Solution
Factor:
$$(x-2)(x-3)>0.$$
Critical points: $2$ and $3$. Test intervals.
- $(-\infty,2)$: choose $x=0$, product positive.
- $(2,3)$: choose $x=2.5$, product negative.
- $(3,\infty)$: choose $x=4$, product positive.
Solution:
$$(-\infty,2)\cup(3,\infty).$$
MyOpenMath: Practice: polynomial inequalities (placeholder)
Rational Inequalities
Include denominator restrictions and open circles at undefined points.
Example: Solve a rational inequality
Solve
$$\frac{x-1}{x+2}\le0.$$
Show Solution
Critical values: $x=1$ (zero), $x=-2$ (undefined).
Test intervals:
- $(-\infty,-2)$: choose $x=-3$, expression positive.
- $(-2,1)$: choose $x=0$, expression negative.
- $(1,\infty)$: choose $x=2$, expression positive.
Need values where expression is $\le0$.
Include $( -2,1 )$ and include $x=1$ because it gives zero. Exclude $x=-2$.
Solution:
$$(-2,1].$$