Understanding Rational Inequalities

The video tutorial explains how to solve a rational inequality by finding the zeros of the numerator and denominator, plotting these on a number line, and testing intervals to determine which satisfy the inequality. It also covers expressing solutions using interval notation and verifying them graphically.

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

• Using interval notation

• Plotting the graph of the function

• Factoring the numerator and denominator

• Testing values in the inequality

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

• The position on the number line

• The type of inequality (greater than or equal to)

• The value of the denominator

• The value of the numerator

Why are zeros of the denominator always plotted as open points?

• Because they are greater than zero

• Because they are less than zero

• Because they result in division by zero

• Because they are part of the solution

Which interval is part of the solution for the inequality (x + 2)/(x - 3) > 0?

• The interval to the left of -2

• The interval between -2 and 3

• The interval to the right of 3

• The interval between 0 and 3

How is the solution to the inequality expressed using interval notation?

• (-∞, -2) ∪ (-2, 3)

• (-2, 3)

• (-∞, -2) ∪ (3, ∞)

• (-∞, 3) ∪ (3, ∞)

What does it mean graphically when the function is above the x-axis?

• The function is less than zero

• The function is equal to zero

• The function is greater than zero

• The function is undefined

Why is it important to verify the solution graphically?

• To ensure the solution is correct

• To find the exact values of x

• To avoid using interval notation

• To simplify the inequality

What happens at x = 3 in the graph of the function (x + 2)/(x - 3)?

• The graph is below the x-axis

• There is a vertical asymptote

• The graph is horizontal

• The graph crosses the x-axis

Which of the following x-values would satisfy the inequality (x + 2)/(x - 3) > 0?

• x = 4

• x = 2

• x = 0

• x = -1

What is the significance of the interval (-∞, -2) in the solution?

• It includes all x-values greater than 3

• It includes all x-values between -2 and 3

• It includes all x-values less than -2

• It includes all x-values greater than -2