Rational Inequalities and Graphing Concepts

The video tutorial explains how to solve rational inequalities by factoring the numerator and denominator, finding their zeros, and plotting these zeros on a number line to form intervals. Each interval is tested to determine which satisfy the original inequality. The solution is expressed using interval notation and inequalities. The tutorial also demonstrates how to verify solutions graphically by plotting the function and identifying when it is on or below the x-axis.

What is the first step in solving a rational inequality?

• Expressing the solution in interval notation

• Testing intervals

• Plotting the zeros on a number line

• Factoring the numerator and denominator

When plotting zeros on a number line, what determines if a point is open or closed?

• The value of the numerator

• The number of intervals

• The value of the denominator

• The original inequality

Why must zeros of the denominator always be open points?

• To avoid division by zero

• To include them in the solution

• To simplify the inequality

• To ensure all intervals are tested

Which value is tested in the interval where x = -3?

• x = -3

• x = 7

• x = 3

• x = 0

What is the result when testing x = 0 in the inequality?

• The interval is part of the solution

• The interval is undefined

• The interval is not part of the solution

• The interval is always positive

How is the solution expressed using interval notation?

• From 2 to positive infinity

• From negative infinity to 6

• From negative infinity to -2, union 2 to 6

• From -2 to 2

What does the graph of the function help verify?

• The zeros of the denominator

• The factors of the numerator

• The number of solutions

• The intervals that satisfy the inequality

What is the significance of a slant asymptote in the graph?

• It indicates where the graph crosses the x-axis

• It helps in factoring the inequality

• It shows the end behavior of the graph

• It determines the zeros of the numerator

What does it mean when the graph is on or below the x-axis?

• The y-values are zero or negative

• The y-values are positive

• The graph is increasing

• The graph is undefined

What is a vertical asymptote in the context of this inequality?

• A line that indicates the maximum value of the graph

• A point where the graph crosses the x-axis

• A line where the graph approaches but never touches

• A point where the graph is undefined