Understanding Algebraic Expressions and Inequalities

The video tutorial covers four mathematical problems, each focusing on different concepts. Problem 125 explores whether r/s is an integer by analyzing factors and prime factors. Problem 126 determines the value of z when z^n equals 1, considering nonzero integers and positive numbers. Problem 127 involves simplifying a complex expression to find the value of s. Finally, Problem 128 tests an inequality involving the absolute value of x. Each problem is approached by evaluating given statements to determine sufficiency.

If r and s are positive integers, which statement is sufficient to determine if r/s is an integer?

• s is a multiple of r.

• r is a multiple of s.

• Every prime factor of s is also a prime factor of r.

• Every factor of s is also a factor of r.

What can be concluded if n is a nonzero integer and z^n = 1?

• z must be less than 0.

• z must be 0.

• z must be 1 or -1.

• z must be greater than 1.

If z is greater than 0, what additional information is needed to determine the value of z when z^n = 1?

• z is equal to 0.

• n is an even number.

• z is less than 1.

• n is a nonzero integer.

What is the simplified form of the expression s = (2/n) / (1/x + 2/3x)?

• 5n/6x

• 2x/3n

• 3x/2n

• 6x/5n

If x = 2n, what is the value of s in the expression s = 6x/5n?

• 6/5

• 5/12

• 12/5

• 5/6

If n = 1/2, why is this information insufficient to determine the value of s?

• n is not an integer.

• x and n are equal.

• s is undefined.

• x is not known.

If x is an integer, under what condition is x * |x| < 2x?

• x is less than 0.

• x is equal to 0.

• x is equal to 1.

• x is greater than 0.

If x = -10, is the inequality x * |x| < 2x true?

• Yes, because -100 < -20.

• Yes, because -100 = -20.

• No, because -100 = 20.

• No, because -100 > -20.

What happens if x is less than -2 in the inequality x * |x| < 2x?

• The inequality is undefined.

• The inequality is sometimes true.

• The inequality is never true.

• The inequality is always true.

Why is statement 1 alone insufficient to determine if x * |x| < 2x when x < 0?

• It does not specify the exact value of x.

• It does not specify if x is an integer.

• It does not specify if x is negative.

• It does not specify if x is positive.