The Correct Answer is (B) — When y = 3, the equation becomes 3x + 2x = 5(3) + 5, or 5x = 20. Dividing by 5 gives x = 4.
The Correct Answer is (C) — Subtracting x from both sides gives x + 6 = 12; subtracting 6 from both sides gives x = 6.
The Correct Answer is (C) — This question is most quickly solved with guess-and-check, but we can also look for the function that is maximized by (has its greatest value when given) negative numbers – in this case, the function that includes a negative x. (C) is the only such function, since (D) simplifies to f(x) = -3 + x.
The Correct Answer is (C) — We can break up the expression f(-x) + 2 to make it easier to find the correct graph. First, look at “f(-x)”. Plugging this into the first equation gives us f(-x) = -x + b. Since both equations are linear, they can be written in the form y = mx + b, so we know that the slope of the graph we are looking for is negative, since m = -1. Next, look at “+ 2”. Adding 2 means we are shifting the graph in the positive y direction by 2, so we need to find the graph with a negative slope, shifted up by 2 relative to the original graph. (C) is the only graph that satisfies these criteria.
The Correct Answer is (B) — We can represent the sum of three consecutive even integers as n + (n + 2) + (n + 4), or 3n + 6. Since this sum must be greater than 20, we can write 3n + 6 > 20. Subtracting 6 from both sides gives 3n > 14. Since 14 is not divisible by 3, we can try the closest even numbers that are: 12 and 18. These give us n = 4 and n = 6. When n = 4, 3n = 12, which is not greater than 14, so n must be 6.
The Correct Answer is (B) — We can take the square root of both sides to get (x – 1) = 2. Adding 1 to both sides gives x = 3. (Since the original term was squared, -1 is another possible value for x, but only positive numbers are given as answer choices).
The Correct Answer is (C) — We can write this equation as f(5) – f(5 + 1) = 2(5) – 5, or f(5) – f(6) = 5. Since we know that f(6) is 12, we can write f(5) – 12 = 5. Adding 12 to both sides gives f(5) = 17.
The Correct Answer is (C) — Every year, the amount of money increases by 5%. This implies an exponential relationship, so we can eliminate (A) and (B). (D) cannot be correct, since when t = 2, d(t) = 1200 + (0.05)(12002) = 1200 + (0.05)(1440000) = 73200, which is far greater than the given value of 1323.
The Correct Answer is (A) — Let x represent the number of molecules of molecule A and y represent the number of molecules of molecule B. Since there are 12 molecules total, x + y = 12. Since it takes 2 milliseconds to decompose molecule A, the time to decompose molecule A can be written as 2x. Similarly, the time to decompose molecule B can be written as y. So 2x + y = 16.
The Correct Answer is (C) — x-2 is the same as , so we can write . Multiplying both sides by x2 gives x3 = 1. x cannot be -1 (-13 = -1), 0 (03 = 0), or 2 (23 = 8), but 13 = 1.
The Correct Answer is (B) — The lowest point of this graph will be at the x value at the midpoint between its x-intercepts, which occur when x = -2 and x = 4. The midpoint of -2 and 4 is 1.
The Correct Answer is (C) — At every interval, the population of beetles doubles. This implies an exponential relationship, which eliminates (A) and (B), which are quadratic. (D) does not work when t = 1 (since 20(0.5)1+1 = 20(0.25) = 5).
The Correct Answer is (C) — Every number in the y column is the square of the number 3 greater than the number in the x column. We can represent this relationship as y = (x + 3)2. Expanding by the FOIL method gives us y = x2 + 6x + 9.
The Correct Answer is (36) — We can divide both sides by 4 to get , then square both sides to get x = 36.
The Correct Answer is (1/9) — We can rewrite this equation as , or . Taking the reciprocal gives , and multiplying by 3 gives x3 = 27, so x = 3. Plugging in to x-2 gives 3-2 or , which is the same as 1/9.
The Correct Answer is (4) — The area of a triangle is ½bh. The height is 4, since the triangle extends from a height of y = 0 to the line y = 4. The base is the x-coordinate of the point where the two lines intersect, that is, the point at which 2x = 4. Since x = 2, b = 2, so the area of the triangle is ½(2)(4) = 4.
The Correct Answer is (16) — We can represent Jane’s age now as j and Mark’s age now as m. So 10 years ago, Jane’s age was j – 10. Since at that time she was twice as old as Mark is now, we can write j – 10 = 2m. If in two years Jane is three times as old as Mark is now, we can write j + 2 = 3m. To find out how old Mark will be in 4 years, we need to find m + 4. We can rewrite our first equation as j = 2m + 10. Plugging this into our second equation gives us (2m + 10) + 2 = 3m. Simplifying gives us 12 = m. This means that Mark is 12 years old, so in 4 years he will be 16.