Rational Inequalities and Interval Notation

This video tutorial focuses on solving rational inequalities. It begins with an example where the inequality is greater than or equal to zero, using a number line to determine the solution in interval notation and inequalities. The second example involves a less than zero inequality, highlighting the importance of open circles and negative regions. The final example addresses a more complex inequality, requiring common denominators and further analysis. The video concludes with a summary of the solutions and encourages viewers to explore additional resources.

What is the first step in solving the rational inequality x - 3 divided by x + 2 is greater than or equal to 0?

• Add 3 to both sides

• Multiply both sides by the denominator

• Set the numerator equal to zero

• Set the denominator equal to zero

How do you represent the solution of the inequality x - 3 divided by x + 2 is greater than or equal to 0 in interval notation?

• (-∞, -2] ∪ (3, ∞)

• (-∞, -2) ∪ [3, ∞)

• (-∞, -2] ∪ [3, ∞)

• (-∞, -2) ∪ (3, ∞)

In the second example, why are all points of interest marked with open circles?

• Because the inequality is less than or equal to zero

• Because the inequality is greater than or equal to zero

• Because the inequality is greater than zero

• Because the inequality is less than zero

What is the interval notation for the solution of the inequality x - 4 times x + 1 divided by x - 3 is less than zero?

• (-∞, 1) ∪ (3, 4)

• (-∞, -1) ∪ (4, ∞)

• (-∞, -1) ∪ (3, ∞)

• (-∞, -1) ∪ (3, 4)

What is the first step in solving the inequality x + 2 over x - 1 is less than or equal to 3?

• Add 3 to both sides

• Divide both sides by 3

• Subtract 3 from both sides

• Multiply both sides by x - 1

After moving 3 to the other side in the third example, what is the next step?

• Multiply by the reciprocal of the denominator

• Set the denominator equal to zero

• Combine the fractions into a single fraction

• Set the numerator equal to zero

What are the points of interest in the third example after setting the inequality to zero?

• 1 and 3

• 2 and 4

• 0 and 3

• 1 and 2.5

In the third example, what is the sign of the expression when x is greater than 2.5?

• Undefined

• Zero

• Negative

• Positive

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

• (-∞, 1] ∪ (2.5, ∞)

• (-∞, 1) ∪ (2.5, ∞)

• (-∞, 1] ∪ [2.5, ∞)

• (-∞, 1) ∪ [2.5, ∞)

How can the solution of the third example be expressed using inequalities?

• x ≥ 1 or x < 2.5

• x > 1 or x ≤ 2.5

• x ≤ 1 or x > 2.5

• x < 1 or x ≥ 2.5