Learning Objectives
Solve a rational inequality by combining terms and analysing signs.
Distinguish numerator zeros from excluded denominator zeros.
Retain holes after cancelling common factors.
Explain why multiplying by a denominator of unknown sign can fail.
Prerequisites: quadratic sign analysis and rational-expression domains. All practice questions in this lesson are original, not official TMUA questions.
What This Lesson Is About
A rational inequality compares an algebraic fraction with another quantity. Its solution depends on the signs of both numerator and denominator, and any input that makes an original denominator zero must be excluded.
This lesson extends sign analysis from products to quotients. We first form one fraction compared with zero, then examine why ordinary cross-multiplication can fail, why cancelled factors leave excluded inputs, and how squared factors affect the sign pattern.
1. Preparing a Rational Inequality for Sign Analysis
A quotient can be positive when both parts share a sign and negative when their signs differ. To apply this rule systematically, identify the allowed inputs and rewrite the comparison against zero.
1.1 Distinguishing numerator zeros from excluded denominator zeros
A rational expression can change sign when its numerator passes through zero or its denominator passes through zero. The latter points are not allowed inputs. Record all original denominator restrictions before combining or cancelling.
1.2 Combining fractions and dividing the number line into intervals
Move every term to one side and combine into a single fraction. Factor numerator and denominator. List their real zeros in order and determine the sign on each intervening interval.
Worked Example: A Single Fraction
Question: Solve (x − 1)/(x + 2) ≥ 0.
Method: Find the numerator zero and denominator exclusion, then compare their signs on each interval and inspect equality at the numerator zero.
The input −2 is excluded and the numerator vanishes at 1. For x < −2, both numerator and denominator are negative, giving a positive quotient. For −2 < x < 1 the quotient is negative, and for x > 1 it is positive.
The solution is x < −2 or x ≥ 1. Include 1 because zero satisfies ≥ 0, but exclude −2 because the expression is undefined there.
2. Understanding Why Denominator Signs Matter
Clearing a denominator multiplies the inequality by an expression. If that expression can be negative, the required comparison direction depends on the input.
2.1 Choosing sign cases or a sign chart
Multiplying an inequality by x is order-preserving only if x > 0 and order-reversing only if x < 0. If its sign is unknown, either split into cases or use a sign chart. Clearing denominators as if the problem were an equation is not valid.
2.2 Using a squared denominator on the allowed domain
An alternative is multiplying by a squared denominator, which is positive on the allowed domain. This preserves the inequality but may raise the degree, so sign analysis is often simpler.
Worked Example: A Misleading Cross-Multiplication
Question: Solve 1/x > 1.
Method: Subtract 1 and combine into one fraction, then analyse signs without multiplying by an unknown-sign input.
Exclude x = 0. Subtract 1: (1 − x)/x > 0. This quotient is negative for x < 0, positive for 0 < x < 1, and negative for x > 1.
Thus 0 < x < 1. Multiplying blindly by x would give only x < 1 and wrongly include negative inputs; for instance, x = −1 makes −1 > 1 false.
3. Keeping Excluded Inputs After Cancellation
Cancelling a common factor can simplify the sign calculation. It does not change which inputs were permitted in the original fraction.
3.1 Removing a cancelled denominator zero from the solution
A common factor can be cancelled for allowed inputs, but any original denominator zero remains excluded. It may lie within an otherwise valid interval. Split that interval or state the exclusion explicitly.
3.2 Checking that a proposed zero is actually defined
Zero is a permitted output in a non-strict inequality only where the original expression is defined. A cancelled denominator point is never rescued by numerator cancellation.
Worked Example: One Missing Point in the Solution
Question: Solve (x² − 1)/(x − 1) > 0.
Method: Record the excluded input before cancellation, solve the simplified inequality, and remove that input from the final set.
The domain is x ≠ 1. On that domain the quotient equals x + 1, so positivity requires x > −1.
The solution is x > −1 with x ≠ 1, equivalently −1 < x < 1 or x > 1. At x = 1 the original is undefined, despite the simplified value being positive.
4. Analysing Repeated Factors
A sign chart must reflect the powers of its factors. A squared factor behaves differently from a single linear factor, so signs need not alternate at every boundary.
4.1 Separating the sign effect from the endpoint restriction
A squared nonzero factor is positive on both sides of its zero, so it does not reverse the sign there. A squared denominator still excludes its zero. A squared numerator is zero there if the denominator is nonzero, so equality may add an isolated point.
4.2 Testing intervals and inspecting each boundary
Work factor by factor instead of assuming the sign alternates at every listed point. After combining fractions, test each interval and inspect every boundary separately.
Worked Example: A Square in the Denominator
Question: Solve (x − 2)/(x + 1)² ≤ 0.
Method: Use the positivity of a nonzero square to reduce the sign test to the numerator, while preserving the denominator exclusion.
For x ≠ −1, the denominator is positive. Therefore the quotient has the sign of x − 2, and it is nonpositive exactly when x ≤ 2.
Keep the domain exclusion: x ≤ 2 with x ≠ −1. The zero at 2 is allowed, while the denominator zero at −1 is not.
Common Mistakes
Do not cross-multiply an inequality by an unknown-sign expression without cases.
Combine the difference of the two sides before making a sign chart.
Zeros of a denominator are always excluded, even for ≤ or ≥.
Signs do not necessarily alternate at squared factors.
5. Practice
Record original denominator exclusions before every calculation. Form one fraction compared with zero, determine its interval signs, then include only those boundary points where the original expression is defined and the comparison allows equality.
1. Solve (x + 3)/(x − 2) < 0.
2. Solve 2/(x − 1) ≤ 1.
3. Solve (x − 1)²/(x + 2) ≥ 0.
4. Solve (x² − 4)/(x − 2) ≤ 0.
5. Solve 1/x ≥ 1/(x + 1).
6. Solve (x − 2)/(x² − 1) > 0.
Worked Solutions
1. Finding opposite-sign numerator and denominator intervals
The quotient is negative between −3 and 2. Both endpoints are excluded, so −3 < x < 2. At 0 it equals −3/2.
2. Moving a constant into a single fraction
Move 1 left: (3 − x)/(x − 1) ≤ 0. This holds for x < 1 or x ≥ 3. The input 1 is excluded; x = 3 gives equality.
3. Accounting for a squared numerator
The squared numerator is positive except at 1. For x > −2 the denominator is positive and the quotient is nonnegative, including 1. For x < −2 it is negative. Thus x > −2.
4. Preserving the domain through cancellation
Exclude 2. Cancellation gives x + 2 ≤ 0, so x ≤ −2. This interval already avoids 2; at −2 the original is 0/(−4) = 0.
5. Comparing two fractions using their difference
The difference is 1/[x(x + 1)], with x ≠ 0, −1. It is positive when the two factors share a sign, so x < −1 or x > 0. It is never zero, so ≥ and > give the same set here.
6. Testing four intervals separated by three critical inputs
Critical inputs are −1, 1 and 2. The signs from left to right are negative, positive, negative, positive. Thus −1 < x < 1 or x > 2. The poles ±1 are excluded, as is numerator zero 2 for this strict inequality.
6. Summary
- Record the domain, combine to one fraction and factor.
- Determine the sign on every interval separated by zeros or undefined points.
- Distinguish equality at a numerator zero from an excluded denominator zero.
- Keep cancelled holes and repeated-factor behaviour in the final solution.