In rational inequalities, an interval is a range of values for which the inequality holds true. These intervals help describe where the expression is positive, negative, or undefined. Here's an outline of intervals in the context of rational inequalities:

Rational Inequality

A rational inequality is an inequality involving a rational function (a ratio of two polynomials). For example: $\frac{f(x)}{g(x)} > 0$ Where $f(x)$ and $g(x)$ are polynomials.

Steps to Solve Rational Inequalities Using Intervals

1. Find critical points: These are the values of $x$ where the rational expression is either equal to zero (where $f(x) = 0$) or undefined (where $g(x) = 0$).

2. Divide the number line into intervals: Use the critical points to divide the real number line into intervals. Each interval represents a possible range of solutions.

3. Test each interval: Pick a test point from each interval and substitute it into the rational inequality to determine whether the inequality is true (positive or negative) in that interval.

4. Check endpoints: Determine whether the inequality includes equality (e.g., $\geq$ or $\leq$) and consider whether the critical points should be included in the solution.

Example:

Solve $\frac{x+2}{x-3} \geq 0$.

1. Critical points: Set $x+2 = 0$ and $x-3 = 0$. These give $x = -2$ and $x = 3$.

2. Intervals: The critical points divide the number line into three intervals:

   • $(-\infty, -2)$
   • $(-2, 3)$
   • $(3, \infty)$

3. Test each interval:

   • For $x = -3$ (in $(-\infty, -2)$), $\frac{-3+2}{-3-3} = \frac{-1}{-6} > 0$.
   • For $x = 0$ (in $(-2, 3)$), $\frac{0+2}{0-3} = \frac{2}{-3} < 0$.
   • For $x = 4$ (in $(3, \infty)$), $\frac{4+2}{4-3} = \frac{6}{1} > 0$.

4. Check endpoints: Since the inequality is $\geq 0$, we include $x = -2$ (because it makes the numerator 0) but exclude $x = 3$ (because the denominator becomes 0, making the expression undefined).

The solution is $(-\infty, -2] \cup (3, \infty)$.

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Would you like more details or have any questions? Here are five related questions to expand this topic:

1. How do you handle inequalities involving more complicated rational functions?
2. What happens when a rational function has higher-degree polynomials?
3. How do you interpret non-strict inequalities (e.g., $\geq$ or $\leq$) in rational inequalities?
4. What if both the numerator and denominator are zero at the same critical point?
5. Can rational inequalities have no solution?

Tip: Always watch out for points where the denominator becomes zero; these are critical for defining where the function is undefined!