Brief Summary
This video explains rational expressions, equations, inequalities, and functions. It begins by defining rational expressions as ratios of two polynomials and then differentiates between rational equations (which contain an equals sign), rational inequalities (which involve inequality symbols), and rational functions (expressed in the form of f(x) or y). The video includes examples to help viewers identify each type and concludes with practice exercises.
- Rational expressions are ratios of two polynomials.
- Rational equations contain an equals sign.
- Rational inequalities involve inequality symbols.
- Rational functions are expressed in the form of f(x) or y.
Rational Expressions
A rational expression is defined as an expression that can be written as a ratio of two polynomials. Examples include 2/x, (x² + 2x + 3)/(x + 1), and 5/(x - 3). To determine if an algebraic expression is rational, both the numerator and denominator must be polynomials. For instance, (x² + 3x + 2)/(x + 4) and 1/(3x²) are rational expressions because both their numerators and denominators are polynomials. However, √(x + 1)/(x³ - 1) and x⁻² - 5/(x³ - 1) are not rational expressions because their numerators are not polynomials (due to the square root and negative exponent, respectively). The expression 1/x + 2/(x - 2), which simplifies to (x - 2 + 2x) / (x(x - 2)), is also a rational expression because its simplified form is a ratio of two polynomials.
Rational Equations, Inequalities, and Functions
A rational equation is an equation that involves rational expressions, characterized by the presence of an equals sign. An example is 5/x - 3/(2x) = 1/5. A rational inequality involves an inequality symbol (such as >, <, ≥, or ≤) with rational expressions; for example, 5/(x - 3) ≤ 2/x. A rational function is a function in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions and q(x) is not equal to zero. An example is f(x) = (x² + 2x + 3)/(x + 1). Rational equations and inequalities can be solved for x values that satisfy the equation or inequality, while a rational function expresses a relationship between two variables, x and y, and can be represented by a table of values or a graph.
Practice Exercises
Several examples are provided to differentiate between rational functions, equations, inequalities, and other expressions. The expression 2 + x/(x - 1) = 8/7 is identified as a rational equation due to the presence of the equals sign. The expression x > √(x + 2) is categorized as "none of these" because it does not involve a ratio of two polynomials. The expression f(x) = (6 - x + 3)/(x² - 5) is identified as a rational function because it is in the form of f(x). The expression 2x ≥ 7/(x + 4) is a rational inequality because it contains a "greater than or equal to" symbol. Lastly, x/2 = 4/(x³ + 9) is a rational equation because it contains an equals sign.
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