Solving Inequalities with Rational Functions

The video tutorial focuses on solving inequalities involving rational functions by testing sections. It begins with an introduction to the topic, followed by identifying critical values where the graph changes from positive to negative or vice versa. The instructor demonstrates how to test sections on a number line to determine solution intervals. Several example problems are solved to reinforce the method, and the video concludes with a summary of the steps and key points.

What is the first step in solving an inequality involving rational functions?

• Plot the critical values on a number line

• Set one side of the inequality to zero

• Find the critical values

• Factor the numerator and denominator

What are critical values in the context of solving inequalities?

• Points where the graph is undefined

• Points where the graph changes from positive to negative or vice versa

• Points where the graph intersects the y-axis

• Points where the graph has a horizontal asymptote

Why must zeros in the denominator be represented as open circles on the number line?

• Because they represent points where the function is negative

• Because they represent points where the function is positive

• Because they represent points where the function is undefined

• Because they represent points where the function is zero

When testing sections on the number line, what should you focus on?

• The exact values of the function

• The sign (positive or negative) of the function

• The slope of the function

• The intercepts of the function

In the example, what is the result of testing the section between -2 and 1?

• Positive

• Undefined

• Zero

• Negative

What is the interval notation for the solution of the inequality in the first example?

• (-∞, -2] ∪ [1, 2)

• (-2, 1) ∪ [2, ∞)

• (-2, 1] ∪ (2, ∞)

• (-∞, -2) ∪ (1, 2)

What is the purpose of factoring the numerator and denominator in solving inequalities?

• To find the intercepts

• To determine the slope

• To find the critical values

• To simplify the equation

In the second example, what is the critical value obtained from the denominator?

• 0

• 1

• 3/2

• -2/3

What should you do after plotting the critical values on the number line?

• Solve for x

• Find the intercepts

• Test each section to determine where to shade

• Shade the entire number line

What is the final step in solving an inequality involving rational functions?

• Plot the graph

• Solve for y

• Find the intercepts

• Describe the solution using interval notation