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.

Let the length of the edge of the smaller cube be s. The surface area is then 6s². If the length of the edges are tripled, then s is replaced by 3s, making the surface area 6(3s)² = (9)6s², or 9 times larger than the initial surface area.

To solve, use the Pythagorean Theorem. The hypotenuse, c, is related to the legs x and 3x by the equation c² = x²+(3x)², which is equivalent to x²+9x², or 10x². Since c² = 10x², c = \(\sqrt{(10x^{2})}\), or \(\sqrt{10}\)x.

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.

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 ¹⁄₂m 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 + \(\frac{1m}{2}\)

\(=\frac{2m}{2}+\frac{1m}{2}\)

\(=\frac{3m}{2}\)

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.

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.

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 slope-intercept 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.

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.