Understanding Rational Inequalities

The video tutorial explains how to solve a rational inequality involving the expression R(x) = (x-5)/(x-7) ≤ 0. It covers verifying the correct form of the inequality, determining zeros of the numerator and denominator, plotting these zeros on a number line, and testing intervals to find the solution. The solution is expressed using interval notation, indicating the range of x values that satisfy the inequality.

What is the correct form of a rational inequality?

• A single quotient on the right and zero on the left

• Two quotients on both sides of the inequality

• A single quotient on the left and a non-zero number on the right

• A single quotient on the left and zero on the right

How do you find the zero of the numerator in the inequality R(x)?

• Set x plus 5 equal to 0

• Set x plus 7 equal to 0

• Set x minus 7 equal to 0

• Set x minus 5 equal to 0

What type of point is used for the zero of the numerator when the inequality is 'less than or equal to'?

• No point

• Closed point

• Open point

• Half-open point

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

• Because they are part of the solution

• Because they are less than the numerator

• Because division by zero is undefined

• Because they are greater than the numerator

What is the result of testing the interval when x equals 0?

• The interval is true

• The interval is false

• The interval is undefined

• The interval is partially true

Which test value makes the interval true when x equals 6?

• The interval is partially true

• The interval is undefined

• The interval is true

• The interval is false

What is the solution interval when expressed in interval notation?

• (5, 7)

• [5, 7]

• [5, 7)

• (5, 7]

How is the solution expressed using inequalities?

• x > 5 and x < 7

• x > 5 and x <= 7

• x >= 5 and x <= 7

• x >= 5 and x < 7

What is the significance of using a square bracket in interval notation?

• It indicates the interval is undefined

• It indicates the interval is partially open

• It indicates the interval is closed

• It indicates the interval is open

Why is the interval open on 7 in the solution?

• Because 7 is included in the solution

• Because 7 is a test value

• Because 7 is not included in the solution

• Because 7 is undefined