# Rational Inequalities ## Introduction A **rational inequality** involves a rational expression (fraction with polynomials) and an inequality sign. **Examples:** - $\frac{x + 1}{x - 2} > 0$ - $\frac{x^2 - 4}{x + 3} \leq 0$ - $\frac{2x}{x^2 - 9} \geq 1$ ## Key Differences from Polynomial Inequalities **Critical values come from TWO sources:** 1. **Zeros of the numerator** (where the expression equals 0) 2. **Zeros of the denominator** (vertical asymptotes - where expression is undefined) **IMPORTANT:** Values that make the denominator zero are **NEVER** included in the solution, even with $\leq$ or $\geq$. ## Solution Strategy ### Step-by-Step Process **Step 1: Move everything to one side** Get the inequality in the form $\frac{P(x)}{Q(x)} \diamond 0$ where $\diamond$ is $<$, $>$, $\leq$, or $\geq$. **Warning:** Never multiply both sides by the denominator (you don't know if it's positive or negative). **Step 2: Find critical values** - **Numerator zeros**: Set $P(x) = 0$ and solve - **Denominator zeros**: Set $Q(x) = 0$ and solve **Step 3: Create intervals** The critical values divide the number line into regions. **Step 4: Make a sign chart** Test a point from each interval to determine the sign of the rational expression. **Step 5: Identify solution intervals** - For $> 0$ or $\geq 0$: positive intervals - For $< 0$ or $\leq 0$: negative intervals **Step 6: Check endpoints** - **Include** numerator zeros if using $\leq$ or $\geq$ - **Never include** denominator zeros (always undefined) ## Sign Chart for Rational Expressions For $\frac{(x - a)(x - b)}{(x - c)}$ where $a < b < c$: | Interval | $(x-a)$ | $(x-b)$ | $(x-c)$ | $\frac{(x-a)(x-b)}{(x-c)}$ | |----------|---------|---------|---------|---------------------| | $x < a$ | $-$ | $-$ | $-$ | $-$ | | $a < x < b$ | $+$ | $-$ | $-$ | $+$ | | $b < x < c$ | $+$ | $+$ | $-$ | $-$ | | $x > c$ | $+$ | $+$ | $+$ | $+$ | **Note:** At $x = c$, the expression is undefined (vertical asymptote). ## Common Mistake to Avoid **WRONG:** Multiplying by the denominator without considering its sign **Example of wrong approach:** $$\frac{x + 1}{x - 2} > 0$$ **Wrong:** Multiply by $(x - 2)$ to get $x + 1 > 0$ **Why it's wrong:** If $x - 2 < 0$, multiplying reverses the inequality! **RIGHT:** Use sign analysis on the rational expression as-is. ## Converting to Standard Form If the inequality is not in the form $\frac{P(x)}{Q(x)} \diamond 0$, rearrange: **Example:** $\frac{x}{x - 1} \geq 2$ Move everything to one side: $$\frac{x}{x - 1} - 2 \geq 0$$ Get common denominator: $$\frac{x - 2(x - 1)}{x - 1} \geq 0$$ $$\frac{x - 2x + 2}{x - 1} \geq 0$$ $$\frac{-x + 2}{x - 1} \geq 0$$ Now analyze this rational expression. ## Special Considerations ### Numerator Zeros vs. Denominator Zeros - **Numerator zero**: Expression equals 0 (may be included in solution) - **Denominator zero**: Expression is **undefined** (never in solution) Use different notation: - Open circle ○ for denominator zeros (excluded) - Closed circle ● for numerator zeros with $\leq$ or $\geq$ (included) ### Multiple Factors Apply the same multiplicity rules as with polynomials: - Odd multiplicity: sign changes - Even multiplicity: sign stays the same ### Simplifying First **Be careful:** Canceling common factors can eliminate critical values! **Example:** $\frac{(x - 2)(x + 1)}{x - 2} \geq 0$ If you cancel $(x - 2)$, you get $x + 1 \geq 0$, which gives $x \geq -1$. **But:** $x = 2$ must be **excluded** (makes original denominator 0). **Correct answer:** $[-1, 2) \cup (2, \infty)$ ## Graphical Interpretation The graph of $y = \frac{P(x)}{Q(x)}$ has: - **Zeros** at numerator zeros (crosses or touches x-axis) - **Vertical asymptotes** at denominator zeros Solution to $\frac{P(x)}{Q(x)} > 0$: where graph is above x-axis Solution to $\frac{P(x)}{Q(x)} < 0$: where graph is below x-axis