For a positive value of \(x\), which of the following could be the correct ordering of \(\frac{1}{x}\), \(2x\), and \(x^2\)?

I. \(x^2 < 2x < \frac{1}{x}\) II. \(x^2 < \frac{1}{x} < 2x\) III. \(2x < x^2 < \frac{1}{x}\)

If \(p\) and \(q\) are integers such that \(q - 3p > 12\) and \(p - q > 38\), which of the following is the least possible value of \(pq\)?

In an algebra practice session, how many integer values of \(x\) satisfy \(x(x+2)(x+4)(x+6) < 200\)?

In an aptitude practice set, positive integers \(x\) and \(y\) satisfy both inequalities \(2x + y < 10\) and \(x - y > -2\). How many ordered pairs \((x,y)\) are possible?

In an algebra quiz, \(x\) is a positive integer. How many values of \(x\) satisfy the inequality \(\frac{x-2}{x^2-13x+36} \le 0\)?

In a coordinate-grid based algebra problem, how many ordered pairs of integers \((x,y)\) satisfy the inequality \(x^2 + 4y^2 < 100\)?

In an algebra practice question, a real number \(x\) satisfies \((|x|-2)(x+5)<0\). Which of the following must be true?

In an algebra practice question, integers \(x\) and \(y\) satisfy \(x^2 + y^2 < 46\) and \(0 < x \le 5\). Which of the following represents all possible values of \(y\)?

In an algebra practice question, a non-zero real number \(x\) satisfies \(\frac{x}{|x|}must be true?

If \(xy>0\) and \((x-2)(y+1)<0\), then which of the following must be true?

How many different pairs of integers \((a,b)\) exist such that \(2\le a\le 200\) and \(a+b>a^b\)?

How many positive integer values can x take that satisfy the inequality \((x-8)(x-10)(x-12)\cdots(x-100)<0\)?

For a non-zero real number \(t\), suppose \(\frac{t^4}{|t|}<\sqrt{t^2}\). Which of the following statements must always be true?

I. \(t<\pi\)

II. \(t^2<1\)

III. \(t^3>-8\)

A project manager assigns two types of resources, x and y, to a campaign. The performance score of the campaign is given by P = 2x + 5y. The allocation must satisfy the conditions x ≤ y, 3x − y ≥ 1, and 3x + y ≤ 5. What is the maximum possible value of P?

In a data-checking algorithm, two non-zero numbers p and q are used to create a ratio such that 0 < p/q < 1. Which of the following statements must always be true?

I. p²/q² < p/q

II. p²/q > p/q

III. (p + 5)/(q + 5) < 1

In a model-testing exercise, two integer parameters a and b are used to define two quadratic expressions. The first expression 2x² − ax + 2 is always positive for every real value of x, and the second expression x² − bx + 8 is always non-negative for every real value of x. What is the largest possible value of 2a − 6b?

A performance index for a machine setting x is given by x² − 7x − 2. For how many integral values of x is this index greater than −8 and less than 6?

In a campus coding challenge, two positive integer scores p and q must satisfy the conditions 2p - 10 > -q and 3q - 20 < -p. If m is the minimum possible value of p and n is the maximum possible value of q, which of the following pairs represents (m, n)?

How many integral values of \(x\) satisfy the inequality \(|3x-5|\le15\)?

If \(x>-6\) and \(\frac{1}{x-5}+\frac{1}{x+3}<0\), how many positive integral values of \(x\) satisfy the given inequality?

If \(x>-6\) and \(\frac{1}{x-5}+\frac{1}{x+3}<0\), what is the sum of all positive integral values of \(x\) satisfying the given inequality?