1.

The Correct Answer is (D) — Cross-multiply to find x: . Therefore, .

2.

The Correct Answer is (B) — We can cross-multiply, or we can just multiply the left side by 12, which when divided by 6, really amounts to a multiplication by 2: . Now we can collect like terms: . The value of is therefore , which is B.

3.

The Correct Answer is (C) — We immediately recognize that this graph is a square-root function, so that rules out A and B. We also notice that the graph is shifted down from the origin, so that rules out D. Therefore, our answer is C.

4.

The Correct Answer is (C) — To find the correct answer, we need to evaluate the truth values of the statements. They use some challenging notation, and the first step is correctly interpreting the notation. Statements I and II define the domain of \(x\) and range of \(y\). Statement I means "the domain of \(x\) is all real numbers except 1," and statement 2 means "the range of \(y\) is all real numbers except 1." After correctly interpreting the statements, we can evaluate their truth values.

Statement I: We can test whether the domain of \(x\) includes 1 by plugging it in to see if it can lead to any valid solution. That gives us a fraction with the denominator \(x-1\), which would be a zero denominator. There's no division by zero, so this value is undefined, and the domain of \(x\) can't include 1. However, this is the only value for which this fraction will be undefined, so any other real value for \(x\) will be equal to some real value for \(y\). Thus, statement I is true. We can therefore eliminate (B), which says that only statement II is true.

Statement II: Again, we can test this statement by plugging 1 in for \(y\) and seeing if it leads to any valid solution.
\(1 = \frac{4}{x-1}\)
\(x = 5\).
Since there's a valid solution for \(y = 1\), \(1\) is in the range of \(y\), and statement II is false. We can now also eliminate (D) and (A). Since only (C) remains, we don't strictly need to determine whether statement III is true. However, we can check it in order to be more confident about our solution.

Statement III: The \(y\) intercept is the point at which the line crosses the \(y\) axis; in other words, it's the point at which \(x\) = 0. So, let's plug 0 in for \(x\): 
\(y = \frac{4}{x-1}\)
\(y = \frac{4}{0 - 1}\)
\(y = \frac{4}{-1}\)
\(y = -4.\)
The \(y\) intercept of this equation is at the point \([0, -4]\), and statement III is true.

NOTE: this problem was updated between the second and third editions of the SAT Guide. Previously, the correct answer was (D). The answer key in the third edition still lists (D), but the correct answer for the updated problem is (C).

5.

The Correct Answer is (A) — The simple function looks like half of a sideways parabola. The given equation is shifted down by 9 units, which eliminates option B, and flipped across the y-axis, which eliminates C and D. We’re left with A, which a few quick plug-ins can verify is the correct answer. (We could even solve this problem without remembering the translations, and plug in a few values for x: we can see that x cannot be positive, or the value under the square root would be negative and therefore imaginary. So we can only use negative values for x, and we will see that the y-value starts low (at -9) and increases, giving a graph as in A.)

6.

The Correct Answer is (A) — First, we note the asymptotes of the graph: they occur at x = -1 for the black graph, and at x = -3 for the gray graph. The only equation with the proper asymptotes (e.g. the value of x that cannot be a solution to the equation, or that the graph will never cross) is A.

7.

The Correct Answer is (D) — First we must solve for x, then we can use its value in the required expression. To solve for x, we square both sides: changes to . Then we simply collect like terms and solve, so x = 3. Substituting this into gives us , which is 49.

8.

The Correct Answer is (6) — To find the time it takes for the person to experience gastrointestinal issues, we remember from the introduction to this problem that gastrointestinal issues first become noticeable when the person has 64 bacteria in their body. Therefore, we set this equation to equal 64: . We know that , so our answer is 6 minutes.

9.

The Correct Answer is (1.5) — Set the equation to equal 64: . We can re-write 16 in terms of 2 to get . From Question 8, we know that 2^6=64, so we can set these expressions equal to each other, too: . Since the bases are the same, we can eliminate them: . Now we simply solve the equation algebraically to see that . Recall from our answer to Question 8 that the first person felt issues at 6 minutes, so we can take the difference to find that this new person experiences issues 1.5 minutes faster than the first person did.

10.

The Correct Answer is (2012 or 2017) — If f(t) in both equations represents median household income, then we can set the two equations equal to each other to find when the median household incomes of each country will be equivalent: . . Square both sides to eliminate the radical, which gives us . Now we simplify the equation and factor it to get , which means that t = 2 or t = 7, so the two possible answers are 2012 and 2017.