If 8a^{6}b^{3} < 0, then which of the following CANNOT be true?

Solution
In order for 8a^{6}b^{3} to be less than zero, either 8 or a^{6} or b^{3} must be less than zero. However, it is obvious that 8 > 0 and any number taken to an even power is non negative. Thus b^{3}<0 and in order for that to be true, b<0. Of the answer choices, only b>0 CANNOT be true.
In the parallelogram below, lengths are given in inches. What is the area of the parallelogram, in square inches?

Solution
The area of a parallelogram is equal to base × height. In the figure, you can see that the base of the parallelogram is 7 and the height of the parallelogram is 9. Thus, the area of the parallelogram is 9 × 7 = 63.
The volume of a cube is given by the formula s^{3}, where s is the length of a side. If a cube has a volume of 64,and the length of each side is halved, the new cube’s volume will be:

Solution
In order to solve this problem, you must realize that since the volume of a cube is equal to the cube of its sides, multiplying the length of the sides by ^{1}⁄_{2} will have the effect of multiplying the volume by (^{1}⁄_{2})^{3}= ^{1}⁄_{8}. The cube in this problem has a volume of 64, so if you halve the length of each side, new cube’s volume will be 64(^{1}⁄_{8})= 8.
For the area of a square to triple, the new side lengths must be the length of the old sides multiplied by:

Solution
Since the area of a square is equal to the square of its sides, multiplying the sides by √3 will have the effect of multiplying the area by (√3)^{2} = 3.
When measured from a point on the ground that is a certain distance from the base of a cell phone tower, the angle of elevation to the top of the tower is 41°, as shown below. The height of the cell phone tower is 200 feet. What is the distance, in feet, to the cell phone tower?

Solution
What is the solution set of 3a − 2 ≤ 7?

Solution
To find the solution set for 3a − 2 ≤ 7, break it up into two separate inequalities: 3a − 2 ≤ 7 and 3a − 2 ≥ −7. Starting with 3a − 2 ≤ 7, solving for a yields a ≤ 3. With 3a − 2 ≥ −7, solving for a yields a ≥ −^{5}⁄_{3}.Thus a is between −^{5}⁄_{3} and 3 inclusive.
The value of b that will make ^{b}⁄_{3} +2 = ^{1}⁄_{4} a true statement lies between which of the following numbers?

Solution
To simplify calculations, you can multiply the entire equation by 12 to obtain whole numbers and get 4b + 24 = 3.
Subtracting 24 from both sides yields 4b = −21.
Dividing by 4 yields b = −\(\frac{21}{4}\) , which is a little less than −5.
Thus the correct answer will lie between −4 and −6.
Which of the following operations will produce the smallest result when substituted for the blank in the expression:^{2}⁄_{3} ____ − 3?

Solution
In the figure below, tan ϕ = ?

Solution
Tangent is the ratio of the side opposite to the side adjacent to an angle in a right triangle. Drawing a line that passes through (3,3) and is perpendicular to the xaxis creates a right triangle, as shown in the figure (see below).
Because point (3,3) is given, both legs of the right triangle have a length of 3. Thus tan ϕ = ^{3}⁄_{3} = 1.
In the standard (x,y) coordinate plane, which of the following lines goes through (3,4) and is parallel to y = 2x + 2?

Solution
Remember that all parallel lines have the same slope, so a line parallel to y = 2x + 2 will a slope of 2. A quick way to aid you in solving this problem would be to eliminate answer choices that do not have slope 2, so answer choices A and E can be immediately eliminated. Check the point (3,−4) in the remaining answer choices. The only choice that works is y = 2x − 2.