Rational Inequality Quiz

Rational Inequality

• Rational inequalities involve inequalities containing rational expressions.
• To solve rational inequalities, first rewrite the inequality as a single rational expression set to be greater than or less than zero.
• Identify the critical values (values where the numerator or denominator is zero). These values divide the number line into intervals.
• Test a value from each interval in the original inequality to determine if it satisfies the inequality. The intervals containing the solution values are the answer. Include endpoints that satisfy the original inequality.
• When the inequality symbol is '≤' or '≥', include any critical values—that make the expression zero.
• When the inequality symbol is '<' or '>', do not include intervals where the expression is equal to zero (because expressions are not equal to 0 in these conditions).
• Consider the sign of the numerator and denominator separately in each interval to determine the overall sign of the rational expression.

Rational Equations

• Rational equations involve equations containing rational expressions.
• To solve rational equations, first find the least common denominator (LCD) of all the fractions in the equation.
• Multiply both sides of the equation by the LCD to eliminate the fractions.
• Solve the resulting equation.
• Check for extraneous solutions (solutions that make the denominator equal to zero in the original equation). Any solution that makes a denominator zero is extraneous and must be rejected.
• The remaining solutions are the solutions to the rational equation.

Rational Function (Video)

• Rational functions are functions of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials, and q(x) ≠ 0.
• A rational function can have vertical asymptotes where the denominator is zero but the numerator is not zero.
• A rational function can have a horizontal asymptote, which is a horizontal line the graph of the function approaches but never touches. The horizontal asymptote is determined by the degrees of the numerator and denominator polynomials.
• A rational function can have oblique (or slant) asymptotes if the degree of the numerator is exactly one greater than the degree of the denominator. The oblique asymptote is a line found by polynomial long division.
• Graphing rational functions involves careful consideration of asymptotes, intercepts, and behavior near asymptotes.

Table of Values

• A table of values lists input (x) values and corresponding output (y = f(x)) values for a function.
• Creating a table of values is useful for visualizing the behavior and shape of a function, especially when evaluating rational functions or polynomial functions.
• Select a range of x-values for the table to get a decent representation. Try both positive and negative values, and include values near where any important features are occurring (root, asymptotes, etc).

Inverse Function (Video)

• The inverse of a function swaps the input (x) and output (y) values. This can be shown algebraically or graphically and can involve changing the input output values based on the original function.
• To find the inverse of a function, replace f(x) with y, switch x and y, then solve for y.
• The inverse function "undoes" the original function, meaning if f(a) = b, then f⁻¹(b) = a.
• A function must be one-to-one (pass the horizontal line test) to have an inverse function. If a function fails the horizontal line test, it can be restricted to a specific domain to create a new, restricted function with an inverse.
• The graph of the inverse function is a reflection of the graph of the original function across the line y = x.

Test your understanding of rational inequalities with this quiz. You'll solve inequalities involving rational expressions and learn to determine critical values and intervals. This quiz is essential for mastering the concept of rational expressions in algebra.