Review Questions of Algebra

　　1. Solve for x in the equation

　　(A)x = –3

　　(B)x = 0

　　(C)x = 6

　　(D)x = 12

　　(D)x = 24

　　2. Solve for x in the following inequality:　> 3

　　(A)x < –13

　　(B)x ≥ –13

　　(C)x ≤ –5

　　(D)x < –5

　　(D)x > –5

　　3. Solve for x in the following equation: 3 + |2x – 7| = x + 2

　　(A)x = –6

　　(B)x = 8

　　(C)x = {, 8}

　　(D)x = {6, 8}

　　(D)x = {, 6}

　　4. If x = 2y – 4 and y = 7s + 1, what is the value of s in terms of x?

　　(A)s =

　　(B)s =

　　(C)s =

　　(D)s =　+ 2 +

　　(D)s =

　　5. Jim and Ryan run laps around a track which is 1 /4 of a mile long. If Jim runs h laps per hour, Ryan runs half as fast as Jim, and Jim starts running half an hour before Ryan, how many miles farther than Ryan has Jim run two hours after Jim starts?

　　(A)

　　(B)

　　(C)

　　(D)

　　(D)

　　6. Ken buys a shirt on a 30% sale from its original price of 10 dollars. He then resells it at a 60% increase from the price at which he bought it. How much does Ken sell the shirt for?

　　(A)$4.20

　　(B)$11.20

　　(C)$13.00

　　(D)$13.30

　　(D)$14.20

　　7. A train leaves the station traveling due east at a rate of 45 miles per hour. A second train leaves a different station an hour later, traveling due west on the same track, going 60 miles per hour. If the stations are 255 miles apart, how many miles from the halfway point between the stations will the trains collide?

　　(A)0

　　(B)5

　　(C)7.5

　　(D)12.5

　　(D)25

　　8. A snowman weighing 250 pounds begins to melt in the spring. It loses 4% of its weight every day. How many full days will pass before it weighs less than 100 pounds?

　　(A)17

　　(B)22

　　(C)23

　　(D)27

　　(D)45

　　9. What is (a + b + 3c)(2a + 3b + 4c)?

　　(A)3a + 4b + 7c

　　(B)2a2 + 3b2 + 12c2

　　(C)2a2 + 10ac + 12c2

　　(D)2a2 + 3ab + 10ac + 9bc + 12c2

　　(D)2a2 + 3b2 + 12c2 + 5ab + 10ac + 13bc

　　10. Solve for x in the following equations: 3x2 + 24x – 27 = 0

　　(A)x = 1

　　(B)x = –9

　　(C)x = {1, –9}

　　(D)x = {–1, 9}

　　(D)x = {3, 27}

　　Explanations

　　1.　　　D

　　To solve this problem, just isolate the variable:

　　2.　　　D

　　Solving for a variable in an inequality is similar to solving for a variable in a normal equation, with one big difference: you isolate the variable. But remember that if you multiply or divide an inequality by a negative number, you must reverse the direction of the inequality symbol:

　　3.　　　E

　　First, isolate the expression within the absolute value brackets:

　　Then divide the equation into two equations. In this first case, the expression within absolute value brackets is positive:

　　Next, let the expression within absolute value brackets be negative:

　　The two solutions are x = {8 3, 6}.

　　4.　　　B

　　First, solve for s in terms of y:

　　Next, solve for y in terms of x:

　　Finally, substitute this value for y in the equation with s:

　　5.　　　A

　　This is a rate question, and we are given the input and rate in order to find the output. So, we can plug the known values into the rate formula:

　　Jim runs h laps per hour for 2 hours, so his total distance traveled is 2h laps, which equals 2h 4 miles. Ryan runs .5h laps per hour for 1.5 hours, so in total he runs 0.75h laps. This is equal to .75h 4 miles. The difference between the distance traveled by Jim and the distance traveled by Ryan is 2h 4 – .75h 4 = 1.25h 4 miles.

　　6.　　　B

　　This is a double percent-change problem, and so we perform each percent change one by one. First, Ken bought the shirt at a discount of 30%. The price at which he paid was:

　　He then sold it for 60% more than he paid:

　　He sold the shirt for $11.20.

　　7.　　　C

　　The toughest part of this rate problem is translating the word problem into an equation. The point at which the trains will collide is the point at which their combined distance traveled is 255 miles. Using this fact and the rates at which the trains travel, we can find out when the collision occurs, in relation to when the trains left their respective stations. Finally, from this newly calculated information, we can find where the collision occurred. Here is the rate formula we’ll be using:

　　Let x represent the number of hours before the trains collide. We then have the equation:

　　This equation explains the situation before the collision: that the train going 45 miles per hour traveled for x hours and the train traveling 60 miles per hour traveled for x – 1 hours. Their combined distance traveled is 255. Now solve the equation for x:

　　3 hours pass before the trains collide. From this, we know that the collision happened 3 45 = 135 miles from the western station, and 2　60 = 120 miles from the eastern station. The halfway point between the stations is 255 2 = 127.5 miles from either station, so it happened 135 – 127.5 = 7.5 miles from the halfway point between the stations.

　　8.　　　C

　　This problem fits the classic exponential decay model. So we plug the given information into the formula:

　　Then we solve:

　　Thus, it takes approximately 22.5 days to reach the 100 pound mark, or, as the question asked, 23 full days.

　　9.　　　E

　　To multiply polynomials two at a time, just distribute the terms of one polynomial into the other one individually:

　　10.　　　C

　　The equation given is in the form of a quadratic equation ax2 + bx + c = 0, so you can use either the reverse FOIL or the quadratic formula to solve for the roots. Before doing either of those things, first factor out 3 from the equation:

　　Factoring takes less time than working out the quadratic formula, so check to see if factoring is possible. It is, and you get:

　　The solution set for x is {1, –9}.