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1.8 · Definition
What is a rational function? What are its zeros? When is it undefined?
p(x)/h(x) — zeros when p=0, undefined when h=0
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1.8 · Sign Chart Rule
When solving a rational inequality, what TWO types of values must appear on the sign chart?
Zeros AND undefined values
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1.8 · Endpoint Rule
For a rational inequality, when do you include vs exclude endpoints in the solution?
Think: zeros vs VAs, and < vs <=
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1.8 · Even Multiplicity
What happens to the sign at a zero or VA with EVEN multiplicity?
Same as polynomial multiplicity rule
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1.8 · Example 1
Solve (x−2)/[(x+6)(x−3)] >= 0. Answer in interval notation.
VAs: x=−6, x=3 · Zero: x=2
Sign: −, +, −, + at those four intervals
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1.8 · Example 3
Solve 2/(x−3) > 0. Answer in interval notation.
Constant positive numerator — what matters?
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1.8 · Example 5
Solve (x−1)(x+2)^2/(x−2) >= 0. What is special about x=−2?
Even multiplicity zero — no sign change
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1.8 · Example 6
Solve 1/(x−1)^2 <= 0. What is the answer?
Always positive expression — is there a solution?
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✓ Sign Chart Critical Values
BOTH zeros (num=0) AND undefined values (den=0)
Missing either type gives wrong signs on intervals. VAs are boundaries too — just never included in the solution.
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✓ Rational Function Definition
f(x) = p(x)/h(x) · Zeros: p(x)=0 · Undefined: h(x)=0
A ratio of two polynomials. Zeros make f(x)=0. Undefined values create vertical asymptotes or holes.
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✓ Even Multiplicity — No Sign Change
Sign stays SAME on both sides of the critical value
Even multiplicity (zero or VA) means the expression does NOT change sign at that point.
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✓ Endpoint Inclusion Rules
Zeros: include for <=/>= (closed) · VAs: NEVER include
Zeros (num=0): f=0 there, include for <= or >=. VAs (den=0): f undefined, always exclude with open bracket.
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✓ Example 3 — 2/(x−3) > 0
Answer: (3, inf)
Numerator 2 is always positive. Sign follows denominator: negative for x<3, positive for x>3. VA at x=3 excluded.
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✓ Example 1 — Sign Chart Result
Answer: (−6, 2] U (3, inf)
VAs: x=−6, x=3 (open). Zero: x=2 (closed). Signs: −,+,−,+. Want >=0: (−6,2] and (3,inf).
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✓ Example 6 — 1/(x−1)2 <= 0
Answer: No solution
1 > 0 always. (x−1)^2 > 0 for all x not equal to 1. Expression is ALWAYS positive. Never <= 0.
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✓ Example 5 — (x−1)(x+2)2/(x−2) >= 0
Answer: (−inf, 1] U (2, inf)
x=−2 has even mult zero — no sign change. Signs: +,+,−,+. Want >=0: (−inf,1] and (2,inf). VA at x=2 excluded.