It's a good idea to sketch the situation that's described in this question:

Put much more simply, what the question is asking is: At what point does cost equal revenue? In algebraic terms, for what value of x does C equal R?

The statement f(x) = can be read, "f of x equals 2 when x does not equal 13, and f of x equals 4 when x equals 13." Since neither of the values you're given equals 13, the function will always come out to 2. f(15) - f(14) = 2 - 2 = 0. The correct answer is (B).

If you add the equations as presented, the y's drop out:

This answer can be found by finding both roots and adding them together.

It's helpful to draw this one. The wire has a slope of , meaning that it rises 2 feet for every 5 feet it runs. Since its total rise is 48, or (2 × 24), its total run must be 120, or (5 × 24). Don't pick answer choice (B), though. The question asks for the length of the wire, not the distance between the anchor and antenna. The wire's length is the hypotenuse of a right triangle with legs 48 and 120. The Pythagorean theorem will tell you that its length is 129.24.

(D)g(2) = 32 = 9. f(g(2)) = f(9) = 31.

E When you're given two equations extremely similar in form, you're probably looking at classic ETS-style simultaneous equations. The best way to solve these? Rack 'em, stack 'em, add or subtract 'em! (Isn't that satisfying?) In this case, adding the two equations cancels out the b term, leaving you with the equation 8a = 16, so a = 2.

E Plug In x = 1,000. Then Rodney has earned 4 × 1,000 = $4,000 and spent $200, so he has made a total of $3,800. That's (E).

When two events are independent, the probability they will both occur is the product of each of their probabilities.

The first couple must compete against 5 couples. The second couple has already faced the first couple, so they have to compete against only four new couples. The third couple has to compete against three new couples, and so on. This means that to make sure everyone competes in a dance-off with everyone else, there have to be 5 + 4 + 3 + 2 + 1 = 15 dance-offs. Multiply 15 by the 3 times this must happen, and you get

(B) Since there are 4 broken transistors, there must be 8 good ones. P (first pick is good) = . Of the remaining 11 transistors, 7 are good, and so P(second pick is good) = . Finally, P(third pick is good) = . Therefore, P(all three are good) = .
Alternative Solution: Note that there are ways to select 3 good transistors.
There are ways to select any 3 transistors. P(3 good ones) =

All the fancy language in this question basically boils down to this: List M and list N each contains 20 elements; each element in list M is larger than the corresponding element in list N.

(C) Since the y- and z-coordinates can have any values, the equation x = 5 is a plane where all points have an x-coordinate of 5. [2.2]

B If logx(yx) = z, then z is the exponent that turns x into yx. If you think about it that way, then it's clear that x raised to the power of z would be yx. If that doesn't make sense to you, then review Chapter 3.
If that doesn't work for you, then it's possible (but a little tricky) to Plug In. Do it this way. Plug 10 in for x so that you're working with a common logarithm, the kind your calculator can compute. Plug In 2 for y and you get: log10(210) = z. This can be written simply as log 1,024 = z. Your calculator can then compute the value of z. z = 3.0103. You can then compute the value of xz. You get 1,024. The only answer choice that equals 1,024 is (B).

B At all points on the y-axis of the coordinate plane, x = 0, so eliminate (D) and (E) right away. Then PITA. In this case, plug each point (x, y) from the answer choices into the equation, to see if it makes the equation true. If the first point in the answer choice works, then try the other point. If they both work, you've found the right answer. If either point fails, cross off that answer choice. Only the two points in (B) fit the given equation. Notice that (D) gives you the x-intercepts.

(D) If you enter imaginary numbers into the calculator, it will do imaginary arithmetic without changing mode. The imaginary unit is 2nd decimal point. Enter the product, and read the solution 23 + 2i .

Sarah can fill her first-period slot with any of the 4 subjects, leaving 3 for the second, 2 for the third, and 1 for the fourth:

(C) Enter the function f into Y1 and the function g into Y2. Evaluate Y1(Y2(2)) to get the correct answer choice C.
An alternative solution is to evaluate g(2) = 5 and f(5) = , and either use your calculator to evaluate or observe that 3 < < 4, indicating 3.61 as the only feasible answer choice. [1.1]

A The simple, grinding way to do this one is to use the distance formula on the answer choices. Any answer choice that produces a distance other than 25 or 26 can be discarded immediately. Only (A) produces distances of 25 and 26 from the two points given.

(A) Sketch a graph of the three vertices. The base is |u | and the altitude is 8. Therefore, the area is 4 |u |. [2.1]

For this question, you need to remember how FOIL works, and that i2 can be replaced by -1.

(C) Let n = 2, f3 = 3; then let n = 3, f4 = 7; and finally let n = 4, f5 = 17. [3.4]

(D) There are 6 choices of color for each of the three candies selected. Therefore, there are 6 × 6 × 6 = 216 color possibilities altogether.

E If you have a graphing calculator, press the Y= key and enter the function. If you check the values of the TABLE, you can find that f(-3.5) and f(2.5) both equal 39. If you don't have a graphing calculator, you can PITA. It may take a while, but you'll get it.

(B)

(D)

For a to be as large as possible, make b as small as possible. If b = 1, then a = 9.

A sphere of radius 5 has volume If that's also the volume of a cube, then an edge of that cube is the cube root of that. Use your calculator:

To solve this problem, sketch an "unfolded" drawing of this solid:

(D) Use the standard window to graph , by moving the cursor all the way left (past Y =) and keying Enter until a "lower triangle" is observed. The shaded portion of the graph will lie in all but Quadrant I.
An alternative solution is to graph the related equation and test points to determine which side of the line contains solutions to the inequality. This will indicate the quadrant that the graph does not enter. [1.2]

To solve an equation with absolute value, consider the two possibilities.

(A) Since g(3) = e3, f (g(3)) = 2 ln e3 + 3. ln e3 = 3. So f(g(3)) = 6 + 3 = 9. [1.4]

Draw a picture of the problem situation:

A The fraction is the reciprocal of ; it's just flipped over. To find the numerical value of , just flip the numerical value of , which is 0.625. Your calculator will tell you that = 1.6.

If Jean can do the whole job in 10 hours, then it takes her to do of the job. If Dan can do the whole job in 12 hours, then it takes him hours to do the other of the job. Thus, together, it takes them hours, or 11 hours and 20 minutes, to paint the house.

B Translate the words into math: = 2. Now peel away. First cube both sides: = 8. Now square both sides: x = 64. PITA works well, too.