Considering all values of a and b for which a + b is at most 9, a is at least 2, and b is at least −2, what is the minimum value of b − a?

Solution
The first step in solving this problem is to rewrite the information in mathematical terms, as follows:
a + b is at most 9 means that a + b ≤ 9
a is at least 2 means that a ≥ 2
b is at least −2 means that b ≥ −2
Given the information above, the value of b − a will be least when b is at its minimum value of −2. In that case, since a + b ≤ 9, then a + (−2) ≤ 9, and a ≤ 11. Therefore, at its minimum, b − a is equivalent to −2 − 11, or −13.
If the edges of a cube are tripled in length to produce a new, larger cube, then the larger cube’s surface area is how many times larger than the smaller cube’s surface area?

Solution
Let the length of the edge of the smaller cube be s. The surface area is then 6s^{2}. If the length of the edges are tripled, then s is replaced by 3s, making the surface area 6(3s)^{2} = (9)6s^{2}, or 9 times larger than the initial surface area.
In the figure below, ΔABC is a right triangle with legs that measure x and 3x inches, respectively. What is the length, in inches, of the hypotenuse?

Solution
To solve, use the Pythagorean Theorem. The hypotenuse, c, is related to the legs x and 3x by the equation c^{2} = x^{2}+(3x)^{2}, which is equivalent to x^{2}+9x^{2}, or 10x^{2}. Since c^{2} = 10x^{2}, c = \(\sqrt{(10x^{2})}\), or \(\sqrt{10}\)x.
Let n equal 3a + 2b − 7. What happens to the value of n if the value of a increases by 2 and the value of b decreases by 1?

Solution
To solve this problem, replace the a and b in 3a + 2b − 7 with a + 2 and b − 1. The result is 3(a + 2) + 2(b − 1) − 7. Distribute to get 3a + 6 + 2b − 2 − 7 = 3a + 2b − 3. Comparing 3a + 2b − 7 and 3a + 2b − 3, it is apparent that the value of n increases by 4 if the value of a increases by 2 and the value of b decreases by 1.
Kate rode her bicycle to visit her grandmother. The trip to Kate’s grandmother’s house was mostly uphill, and took m minutes. On the way home, Kate rode mostly downhill and was able to travel at an average speed twice that of her trip to her grandmother’s house. Which of the following expresses the total number of minutes that Kate bicycled on her entire trip?

Solution
According to the problem, Kate traveled distance d in m minutes on the way to her grandmother’s house, and she traveled distance d in ^{1}⁄_{2}m minutes (because she went twice as fast, it took her half as long) on the way back. The total number of minutes traveled would be equal to the number of minutes Kate traveled to her grandmother’s house and back:
1m + \(\frac{1m}{2}\)
\(=\frac{2m}{2}+\frac{1m}{2}\)
\(=\frac{3m}{2}\)
The City Council has approved the construction of a circular pool in front of City Hall. The area available for the pool is a rectangular region 12 feet by 18 feet, surrounded by a brick wall. If the pool is to be as large as possible within the walled area, and edge of the pool must be at least 2 feet from the wall all around, how many feet long should the radius of the pool be?

Solution
First, draw the picture of the circular pool according to the information given in the problem, where the distance from the edge of the pool to the edge of the long side of the rectangular region is 2 feet. The distance from the edge of the pool to the edge of the sin short side of the rectangular region can be anything greater than 2, but it is not necessary to know this distance to solve the problem:
Now you can determine the diameter of the circular pool. The diameter is the maximum distance from one point on a circle to another (the dashed line). Since the short side of the rectangular region is 12 feet, and the distance from the edge of the circular pool to each edge of the long sides of the rectangular region is set at 2 feet, the diameter of the circle must be 12 feet – 2(2 feet), or 12 feet –4 feet, or 8 feet. The question asks for the radius of the pool, which is ^{1}⁄_{2} of the diameter, or 4.
If cos θ = −^{3}⁄_{5} and ^{π}⁄_{2} < θ < π, then tan θ = ?

Solution
If each element in a data set is multiplied by 3, and each resulting product is then reduced by 4, which of the following expressions gives the mean of the resulting data set in terms of x?

Solution
For the sake of simplicity, let every element in the original set have the value x. If each element in the set is multiplied by 3, and then reduced by 4, each element then has the value 3x − 4. In a set where each value is 3x − 4, the mean is 3x − 4.
What is the slope of a line that is perpendicular to the line determined by the equation 7x + 4y = 11?

Solution
Perpendicular lines have slopes that are opposite reciprocals. To find the slope of a line perpendicular to 7x + 4y = 11, first find the slope by converting the equation to slopeintercept form, then take the opposite reciprocal. To do so, first subtract from both sides to get 4y = −7x + 11. Next, divide both sides by 4 to get y = −\(\frac{7x}{4}\) + \(\frac{11}{4}\). Since the slope in this line is ^{7}⁄_{4}, the slope of a line perpendicular to that is ^{4}⁄_{7}.
Which of the following is the set of all real numbers x such that x − 3 < x − 5?

Solution
To find the real numbers x such that x −3 < x −5, you could subtract x from both sides. The result is −3 < −5, and because that inequality is never true, there is no solution for x. The solution set is the empty set. If you chose an incorrect answer you might have thought that a negative value for x might reverse the inequality, which is not the case.