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 $a < b < c$:

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

Step 1: Factor the numerator $\frac{(x - 2)(x + 2)}{x + 3} \leq 0$

Step 2: Find critical values

Numerator zeros:

• $x - 2 = 0 \Rightarrow x = 2$
• $x + 2 = 0 \Rightarrow x = -2$

Denominator zero:

• $x + 3 = 0 \Rightarrow x = -3$

Critical values: $x = -3, -2, 2$ (in order)

Step 3: Make a sign chart

Step 4: Identify where expression $\leq 0$

We need negative or zero:

• $x < -3$: negative ✓
• $-2 < x < 2$: negative ✓

Step 5: Check endpoints

• At $x = -3$: Undefined (denominator zero), exclude
• At $x = -2$: Expression equals 0, and we have $\leq$, so include
• At $x = 2$: Expression equals 0, and we have $\leq$, so include

Answer: $(-\infty, -3) \cup [-2, 2]$

Verification:

• At $x = -4$: $\frac{(-4)^2-4}{-4+3} = \frac{12}{-1} = -12 < 0$ ✓
• At $x = -2$: $\frac{(-2)^2-4}{-2+3} = \frac{0}{1} = 0$ ✓
• At $x = 0$: $\frac{0-4}{0+3} = \frac{-4}{3} < 0$ ✓
• At $x = 2$: $\frac{4-4}{2+3} = \frac{0}{5} = 0$ ✓

Step 1: Move everything to one side $\frac{2x}{x^2 - 9} - 1 \geq 0$

Step 2: Get common denominator $\frac{2x - (x^2 - 9)}{x^2 - 9} \geq 0$ $\frac{2x - x^2 + 9}{x^2 - 9} \geq 0$ $\frac{-x^2 + 2x + 9}{x^2 - 9} \geq 0$

Step 3: Factor

Numerator: $-x^2 + 2x + 9$

Using quadratic formula: $x = \frac{-2 \pm \sqrt{4 + 36}}{-2} = \frac{-2 \pm \sqrt{40}}{-2} = \frac{-2 \pm 2\sqrt{10}}{-2} = 1 \mp \sqrt{10}$

So numerator zeros are: $x = 1 - \sqrt{10} \approx -2.16$ and $x = 1 + \sqrt{10} \approx 4.16$

Denominator: $x^2 - 9 = (x - 3)(x + 3)$

Denominator zeros: $x = -3, 3$

Critical values (in order): $1 - \sqrt{10}, -3, 3, 1 + \sqrt{10}$

Approximately: $-2.16, -3, 3, 4.16$

Step 4: Make sign chart

Step 5: Identify where expression $\geq 0$

Positive or zero:

• $[1 - \sqrt{10}, -3)$: positive, include left endpoint
• $(3, 1 + \sqrt{10}]$: positive, include right endpoint

Never include $x = -3$ or $x = 3$ (undefined).

Answer: $[1 - \sqrt{10}, -3) \cup (3, 1 + \sqrt{10}]$

Or approximately: $[-2.16, -3) \cup (3, 4.16]$

Verification at $x = 0$ (should be negative): $\frac{2(0)}{0^2 - 9} = \frac{0}{-9} = 0 < 1$ So $\frac{2x}{x^2-9} - 1 = -1 < 0$ ✓