Rational Inequalities and Asymptotes

Professor Dave explains how to graph inequalities, starting with basic concepts and moving to more complex polynomial and rational inequalities. He demonstrates solving inequalities by rearranging terms, using number lines, and testing intervals. The video also covers graphing two-variable inequalities by treating them as equations and shading the solution area. Rational inequalities are addressed by finding intercepts and asymptotes, then sketching the function to determine solution intervals.

What is the main difference between an equation and an inequality?

• An equation is solved graphically, while an inequality is solved algebraically.

• An equation shows equality, while an inequality shows a range of values.

• An equation is always true, while an inequality is always false.

• An equation has variables, while an inequality does not.

When solving the inequality X^2 + 6X < -5, what is the first step?

• Subtract 5 from both sides.

• Add 5 to both sides.

• Multiply both sides by 2.

• Divide both sides by X.

How do you determine which sections of the number line satisfy a polynomial inequality?

• By using a calculator.

• By graphing the inequality.

• By testing points from each section.

• By solving the inequality directly.

What does shading above a line in a graph of an inequality indicate?

• The line itself is not part of the solution.

• The solution includes all points below the line.

• The line is the only solution.

• The solution includes all points above the line.

In the inequality Y > X^2 - 4, what does the graph represent?

• A horizontal line at Y = 4.

• A line shifted down by 4.

• A parabola with solutions above it.

• A circle centered at the origin.

What is the first step in solving a rational inequality like (2X + 2)/(X + 3) > 0?

• Add 3 to both sides.

• Multiply both sides by zero.

• Find the X-intercept by setting the function to zero.

• Subtract 2 from both sides.

How do you find the vertical asymptote of a rational function?

• Set the numerator equal to zero.

• Set the denominator equal to zero.

• Set the function equal to one.

• Set the entire function equal to zero.

What does the union symbol (U) represent in the solution of an inequality?

• The intersection of two sets.

• The exclusion of two intervals.

• The difference between two sets.

• The combination of two intervals.

In the context of rational inequalities, what does it mean when a function is 'greater than zero'?

• The function is above the X-axis.

• The function is equal to zero.

• The function is undefined.

• The function is below the X-axis.

What is the significance of testing points near the asymptote in a rational inequality?

• To calculate the slope of the function.

• To eliminate the asymptote from the graph.

• To find the exact value of the asymptote.

• To determine the behavior of the function around the asymptote.