Pre-Calculus Prep: Solving Rational Inequalities

In pre-calculus, another important topic is solving rational inequalities. Rational inequalities involve rational expressions (fractions) with inequality symbols. The goal is to find the values of the variable that satisfy the inequality.

To solve rational inequalities, we follow these steps:

1. Simplify the rational expression: If the rational expression is not in its simplest form, simplify it by canceling common factors and putting it in the standard form (numerator over denominator).

2. Determine the critical points: The critical points are the values of the variable where the rational expression equals zero or where the inequality symbol changes. To find the critical points, set the numerator equal to zero and solve for the variable. These critical points divide the number line into intervals.

3. Identify the excluded values: Determine any values of the variable that make the denominator of the rational expression equal to zero. These values are called excluded values and must be excluded from the solution set because they would result in undefined expressions.

4. Test the intervals and sign analysis: Choose a test point from each interval and substitute it into the original rational expression. Determine if the expression is positive or negative in that interval. This helps identify which intervals satisfy the original inequality.

5. Express the solution: Combine the intervals that satisfy the original inequality, making sure to exclude any values that make the denominator zero. Express the solution set using interval notation or set-builder notation.

When solving rational inequalities, it is important to consider vertical asymptotes (values that make the denominator zero) and horizontal asymptotes (limits as x approaches positive or negative infinity). These can help determine the behavior of the rational expression and the intervals that satisfy the inequality.

By following these steps, you can determine the values of the variable that satisfy the given rational inequality. It is essential to be cautious when simplifying the expression and analyzing the signs in each interval to accurately identify the solution set.

These videos are designed to review and reteach Precalculus and Collegeboard Pre-CALC AP content. My videos cover functions, polynomials, exponential and logarithmic expressions, trigonometry, parametric equations, polar coordinates, vectors, matrices and systems, conic sections, discrete mathematics, sequences and series; and an introduction to calculus.

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa .
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