A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents held 2 people each, and the rest held 4 people each. Assuming all the tents were filled to capacity and every person got to sleep in a tent, exactly how many of the tents were 2person tents?

Solution
Let x represent the number of 2person tents and let y represent the number of 4person tents.
It is given that the total number of tents was 60 and the total number of people in the group was 202.
This situation can be expressed as a system of two equations, x + y = 60 and 2x + 4y = 202.
The first equation can be rewritten as y = −x + 60.
Substituting −x + 60 for y in the equation 2x + 4y = 202 yields 2x + 4(−x + 60) = 202.
Distributing and combining like terms gives −2x + 240 = 202.
Subtracting 240 from both sides of −2x + 240 = 202 and then dividing both sides by −2 gives x = 19.
Therefore, the number of 2person tents is 19.
Alternate approach: If each of the 60 tents held 4 people, the total number of people that could be accommodated in tents would be 240.
However, the actual number of people who slept in tents was 202.
The difference of 38 accounts for the 2person tents.
Since each of these tents holds 2 people fewer than a 4person tent, ^{38}⁄_{2} = 19 gives the number of 2person tents.
Choice A is incorrect. This choice may result from assuming exactly half of the tents hold 2 people.
If that were true, then the total number of people who slept in tents would be 2(30) + 4(30) = 180; however, the total number of people who slept in tents was 202, not 180.
Choice B is incorrect. If 20 tents were 2person tents, then the remaining 40 tents would be 4person tents.
Since all the tents were filled to capacity, the total number of people who slept in tents would be 2(20) + 4(40) = 40 + 160 = 200; however, the total number of people who slept in tents was 202, not 200.
Choice D is incorrect. If 18 tents were 2person tents, then the remaining 42 tents would be 4person tents.
Since all the tents were filled to capacity, the total number of people who slept in tents would be 2(18) + 4(42) = 36 + 168 = 204; however,the total number of people who slept in tents was 202, not 204.
(x – 6)^{2} + (y + 5)^{2} = 16
In the xyplane, the graph of the equation above is a circle. Point P is on the circle and has coordinates (10, 5). If \(\bar{PQ}\) is a diameter of the circle, what are the coordinates of point Q ?

Solution
The standard form for the equation of a circle is (x – h)^{2} + (y – k)^{2} = r^{2}, where (h, k) are the coordinates of the center and r is the length of the radius.
According to the given equation, the center of the circle is (6, –5). Let (x_{1}, y_{1}) represent the coordinates of point Q.
Since point P (10, –5) and point Q (x_{1}, y_{1}) are the endpoints of a diameter of the circle, the center (6, –5) lies on the diameter, halfway between P and Q.
Therefore, the following relationships hold:
\(\frac{x_{1} + 10}{2} = 6\) and \(\frac{y_{1} + \left(5 \right)}{2} = 5\).
Solving the equations for x_{1} and y_{1}, respectively, yields x_{1} = 2 and y_{1} = −5. Therefore, the coordinates of point Q are (2, –5).
Alternate approach: Since point P(10, −5) on the circle and the center of the circle (6, −5) have the same ycoordinate, it follows that the radius of the circle is 10 – 6 = 4.
In addition, the opposite end of the diameter \(\bar{PQ}\) must have the same ycoordinate as P and be 4 units away from the center.
Hence, the coordinates of point Q must be (2, –5).
Choices B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter \(\bar{PQ}\).
If either of these points were point Q, then \(\bar{PQ}\) would not be the diameter of the circle.
Choice C is incorrect because (6, −5) is the center of the circle and does not lie on the circle.
f(x) = x^{3} – 9x
g(x) = x^{2} – 2x – 3
Which of the following expressions is equivalent to \(\frac{f\left( x \right)}{g\left( x \right)}\), for x > 3?

Solution
Since, x^{3}  9x = x(x + 3)(x  3) and x^{2}  2x  3 = (x + 1)(x  3), the fraction \(\frac{f\left( x \right)}{g\left( x \right)}\) can be written as \(\frac{x \left(x + 3 \right)\left(x  3 \right)}{\left(x + 1 \right)\left(x  3 \right)}\).
It is given that x > 3, so the common factor x 3 is not equal to 0. Therefor, the fraction can be further simplified to \(\frac{x \left(x + 3 \right)}{\left(x + 1 \right)}\).
Choice A is incorrect. The expression \(\frac{1}{x + 1}\) is not equivalent to \(\frac{f\left( x \right)}{g\left( x \right)}\) because at x = 0, \(\frac{1}{x + 1}\) as a value of 1 and \(\frac{f\left( x \right)}{g\left( x \right)}\) has a value of 0.
Choice B is incorrect and results from omitting the factor x in the factorization of f(x).
Choice C is incorrect and may result from incorrectly factoring g(x) as (x + 1)(x + 3) instead of (x + 1)(x – 3).
What is the solution set of the equation above?
\(\sqrt{2x + 6} + 4 = x + 3\)
Solution
Subtracting 4 from both sides of \(\sqrt{2x + 6} + 4 = x + 3\) isolates the radical expression on the left side of the equation as follows: \(\sqrt{2x + 6} = x – 1\).
Squaring both sides of \(\sqrt{2x + 6} = x – 1\) yields 2x + 6 = x^{2} − 2x + 1.
This equation can be rewritten as a quadratic equation in standard form: x^{2} – 4x – 5 = 0.
One way to solve this quadratic equation is to factor the expression x^{2} − 4x – 5 by identifying two numbers with a sum of –4 and a product of –5.
These numbers are –5 and 1. So the quadratic equation can be factored as (x – 5)(x + 1) = 0.
It follows that 5 and –1 are the solutions to the quadratic equation.
However, the solutions must be verified by checking whether 5 and –1 satisfy the original equation, \(\sqrt{2x + 6} + 4 = x + 3\).
When x = –1, the original equation gives \(\sqrt{2 \left(1 \right) + 6} + 4 = \left(1 \right) + 3\), or 6 = 2, which is false.
Therefore, –1 does not satisfy the original equation. When x = 5, the original equation gives \(\sqrt{2 \left(5 \right) + 6} + 4 = 5 + 3\), or 8 = 8, which is true.
Therefore, x = 5 is the only solution to the original equation, and so the solution set is {5}.
Choices A, C, and D are incorrect because each of these sets contains at least one value that results in a false statement when substituted into the given equation.
For instance, in choice D, when 0 is substituted for x into the given equation, the result is \(\sqrt{2 \left(0 \right) + 6} + 4 = \left(0 \right) + 3\) or √6 + 4 = 3.
This is not a true statement, so 0 is not a solution to the given equation.
y > 2x – 1
2x > 5
Which of the following consists of the ycoordinates of all the points that satisfy the system of inequalities above?

Solution
Subtracting the same number from each side of an inequality gives an equivalent inequality.
Hence, subtracting 1 from each side of the inequality 2x > 5 gives 2x − 1 > 4.
So the given system of inequalities is equivalent to the system of inequalities y > 2x − 1 and 2x − 1 > 4, which can be rewritten as y > 2x − 1 > 4.
Using the transitive property of inequalities, it follows that y > 4.
Choice A is incorrect because there are points with a ycoordinate less than 6 that satisfy the given system of inequalities.
For example,(3, 5.5) satisfies both inequalities.
Choice C is incorrect. This may result from solving the inequality 2x > 5 for x, then replacing x with y.
Choice D is incorrect because this inequality allows yvalues that are not the ycoordinate of any point that satisfies both inequalities.
For example, y = 2 is contained in the set y > ^{3}⁄_{2}; however, if 2 is substituted into the first inequality for y, the result is x < ^{3}⁄_{2}.
This cannot be true because the second inequality gives x > ^{5}⁄_{2}.
The width of a rectangular dance floor is w feet. The length of the floor is 6 feet longer than its width. Which of the following expresses the perimeter, in feet, of the dance floor in terms of w ?

Solution
It is given that the width of the dance floor is w feet. The length is 6 feet longer than the width; therefore, the length of the dance floor is w + 6.
So the perimeter is w + w + (w + 6) + (w + 6) = 4w + 12.
Choice A is incorrect because it is the sum of one length and one width, which is only half the perimeter.
Choice C is incorrect and may result from using the formula for the area instead of the formula for the perimeter and making a calculation error.
Choice D is incorrect because this is the area, not the perimeter, of the dance floor.
In the figure above, RT = TU. What is the value of x ?

Solution
Since RT = TU, it follows that ∆RTU is an isosceles triangle with base RU.
Therefore, ∠TRU and ∠TUR are the base angles of an isosceles triangle and are congruent. Let the measures of both ∠TRU and ∠TUR be t°.
According to the triangle sum theorem, the sum of the measures of the three angles of a triangle is 180°.
Therefore, 114° + 2t° = 180°, so t = 33.
Note that ∠TUR is the same angle as ∠SUV. Thus, the measure of ∠SUV is 33°.
According to the triangle exterior angle theorem, an external angle of a triangle is equal to the sum of the opposite interior angles.
Therefore, x° is equal to the sum of the measures of ∠VSU and ∠SUV; that is, 31° + 33° = 64°.
Thus, the value of x is 64.
Choice B is incorrect. This is the measure of ∠STR, but ∠STR is not congruent to ∠SVR.
Choices A and D are incorrect and may result from a calculation error
The formula below is often used by project managers to compute E, the estimated time to complete a job, where O is the shortest completion time, P is the longest completion time, and M is the most likely completion time.
\(E = \frac{O + 4M + P}{6}\)Which of the following correctly gives P in terms of E, O, and M ?

Solution
Multiplying both sides of the equation by 6 results in 6E = O + 4M + P. Then, subtracting O + 4M from both sides of 6E = O + 4M + P gives P = 6E − O – 4M.
Choice B is incorrect. This choice may result from solving for −P instead of for P.
Choice C is incorrect and may result from transposing P with E in the given equation rather than solving for P.
Choice D is incorrect and may result from transposing P with E and changing the sign of E rather than solving for P.
The graph above shows the distance traveled d, in feet, by a product on a conveyor belt m minutes after the product is placed on the belt. Which of the following equations correctly relates d and m ?

Solution
The line passes through the origin. Therefore, this is a relationship of the form d = km, where k is a constant representing the slope of the graph.
To find the value of k, choose a point (m, d) on the graph of the line other than the origin and substitute the values of m and d into the equation.
For example, if the point (2, 4) is chosen, then 4 = k(2), and k = 2. Therefore, the equation of the line is d = 2m.
Choice B is incorrect and may result from calculating the slope of the line as the change in time over the change in distance traveled instead of the change in distance traveled over the change in time.
Choices C and D are incorrect because each of these equations represents a line with a dintercept of 2. However, the graph shows a line with a dintercept of 0.
3x + x + x + x − 3 − 2 = 7 + x + x
In the equation above, what is the value of x ?

Solution
Combining like terms on each side of the given equation yields 6x − 5 = 7 + 2x.
Adding 5 to both sides of 6x − 5 = 7 + 2x and subtracting 2x from both sides yields 4x = 12.
Dividing both sides of 4x = 12 by 4 yields x = 3.
Choices A, B, and C are incorrect because substituting those values into the equation 3x + x + x + x − 3 − 2 = 7 + x + x will result in a false statement.
For example, in choice B, substituting 1 for x in the equation would give 3(1) + 1 + 1 + 1 – 3 – 2 = 7 + 1 + 1, which yields the false statement 1 = 9; therefore, x cannot equal 1.