What are Rational Inequalities in High School Algebra 2?

📚 What are Rational Inequalities?

Rational inequalities are inequalities that involve rational expressions (fractions where the numerator and/or denominator are polynomials). Solving them requires a bit more care than solving regular inequalities because you need to consider the values that make the denominator zero, as these values are not allowed in the solution set (since division by zero is undefined). 🤔

📜 A Little History

The study of inequalities dates back to ancient times, with mathematicians like Diophantus exploring methods to solve various types of inequalities. Rational inequalities, as a specific type, gained prominence with the development of algebraic notation and methods in the 16th and 17th centuries. They became an essential tool for modelling and solving problems in physics, engineering, and economics. 📈

🔑 Key Principles for Solving Rational Inequalities

• ⚖️ Step 1: Rewrite the inequality: Bring all terms to one side, resulting in a single rational expression compared to zero. In other words, manipulate the inequality so it looks like $\frac{P(x)}{Q(x)} > 0$, $\frac{P(x)}{Q(x)} < 0$, $\frac{P(x)}{Q(x)} \geq 0$, or $\frac{P(x)}{Q(x)} \leq 0$.
• 📍 Step 2: Find critical values: Determine the values of $x$ that make the numerator $P(x)$ equal to zero (roots) and the values that make the denominator $Q(x)$ equal to zero (undefined points). These are your critical values.
• 📊 Step 3: Create a sign chart: Draw a number line and mark all the critical values on it. These values divide the number line into intervals. Choose a test value within each interval and plug it into the rational expression $\frac{P(x)}{Q(x)}$. Determine the sign (positive or negative) of the expression in each interval.
• ✅ Step 4: Determine the solution: Based on the original inequality (e.g., $\frac{P(x)}{Q(x)} > 0$), identify the intervals where the rational expression has the correct sign. Remember to consider whether the endpoints (critical values) should be included or excluded based on the inequality symbol (strict inequalities ($<$ or $>$) exclude endpoints; non-strict inequalities ($\leq$ or $\geq$) include endpoints where the numerator is zero, but *never* include endpoints where the denominator is zero).
• ✍️ Step 5: Write the solution: Express the solution in interval notation.

🌍 Real-World Examples

Rational inequalities pop up in various applications. Here are a couple:

• 🧪 Chemical Concentrations: Imagine a chemist needs to maintain the concentration of a substance within a certain range. The concentration might be expressed as a rational function, and the acceptable range defined by a rational inequality.
• 🚗 Production Costs: A factory's cost per unit might decrease as production volume increases, but there's a volume threshold beyond which costs start to rise again due to logistical issues. Modeling this scenario can involve rational functions and inequalities to determine optimal production levels.

✍️ Example 1: Solving a Basic Rational Inequality

Let's solve the inequality $\frac{x-2}{x+3} > 0$.

1. The inequality is already in the correct form.
2. Critical values are $x = 2$ (numerator is zero) and $x = -3$ (denominator is zero).
3. Create a sign chart with intervals $(-\infty, -3)$, $(-3, 2)$, and $(2, \infty)$.
4. Test values: $x = -4$, $x = 0$, $x = 3$.
• For $x = -4$: $\frac{-4-2}{-4+3} = \frac{-6}{-1} = 6 > 0$ (Positive)
• For $x = 0$: $\frac{0-2}{0+3} = \frac{-2}{3} < 0$ (Negative)
• For $x = 3$: $\frac{3-2}{3+3} = \frac{1}{6} > 0$ (Positive)
5. The solution is $(-\infty, -3) \cup (2, \infty)$. Notice that -3 is excluded because the denominator cannot be zero.

💡 Example 2: A More Complex Case

Solve $\frac{2x+1}{x-4} \leq 1$.

1. Rewrite: $\frac{2x+1}{x-4} - 1 \leq 0 \Rightarrow \frac{2x+1 - (x-4)}{x-4} \leq 0 \Rightarrow \frac{x+5}{x-4} \leq 0$.
2. Critical values: $x = -5$ and $x = 4$.
3. Sign chart with intervals $(-\infty, -5)$, $(-5, 4)$, and $(4, \infty)$.
4. Test values: $x = -6$, $x = 0$, $x = 5$.
• For $x = -6$: $\frac{-6+5}{-6-4} = \frac{-1}{-10} = \frac{1}{10} > 0$ (Positive)
• For $x = 0$: $\frac{0+5}{0-4} = \frac{5}{-4} < 0$ (Negative)
• For $x = 5$: $\frac{5+5}{5-4} = \frac{10}{1} = 10 > 0$ (Positive)
5. The solution is $[-5, 4)$. Notice that $-5$ is included because the inequality is non-strict and makes the numerator zero, but $4$ is excluded because it makes the denominator zero.

📝 Conclusion

Rational inequalities might seem daunting at first, but by systematically identifying critical values, creating sign charts, and carefully considering endpoints, you can master them! Remember to always check your solutions and practice consistently. 💪