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1. Find the solution sets for the following: |5x – 7| > 12
Pretend for a moment that the problem was |5x – 7| = 12. This would mean that 5x – 7 = 12 or that 5x – 7 = 12. You would then solve both equations to get two answers for x.
However, the actual problem has a “>” instead of an “=”. This means that 5x – 7 can equal numbers like 13, 14,
15, ... or it can equal numbers like -13, -14, -15....
We have two inequalities we need to solve: 5x – 7 > 12 (the first scenario) and 5x – 7 < -12 (the second
scenario):
5x – 7 > 12
5x > 19
x > 19/5
5x – 7 < -12
5x < -5
x < -1
Therefore, x can be greater than 19/5 OR it can be less than -1.
2. Find the equation of the line satisfying the following:
The line with x-intercept 7 and y-intercept -5.
The x-intercept is the point where the line crosses the x-axis (and therefore y = 0). Similarly, x = 0 at the yintercept. Therefore, the information in the problem tells us two of the points on the line: (7, 0) and (0, -5).
I think the easiest way to solve linear equations is to think of the slope-intercept form of the line and solve for
the slope and the y-intercept (it is especially easy in this case, because they give us the y-intercept directly).
The slope-intercept equation of a line looks like this:
y = mx + b
where m is the slope, and b is the y-intercept.
We already know the y-intercept (it’s -5), so now we need to find the slope.
You calculate the slope by taking two points on the line (which we have), and dividing the difference in the yvalues by the difference in the x-values. So, in this case, the slope is:
(0) – (-5) = 5
(7) – (0) 7
The slope is 5/7.
Plugging the slope and the y-intercept into the equation gives us our answer:
y = (5/7)x – 5
3. Graph the functions L(x) = -3x + 17 and Q(x) = 2x2 + 7x – 22 on the same set of axes.
There are two ways of graphing functions: (1) you can find lots of points that satisfy the function, then plot
those points and connect them, or (2) you can use the equation to find properties of the function (vertex, slope,
intercepts), and use that information to make a graph.
Because it’s easy to find a slope and intercept of a line, we’ll use method 2 to graph L(x), and because it’s more
difficult to find similar information about a parabola, we’ll use method 1 to graph Q(x).
L(x) = -3x + 17
From the equation, we know that the y-intercept is at 17. Put a dot at the point (0, 17). We also know that the
slope is -3. Starting at the point (0, 17), move to the right 1 and down 3 and put another dot there. Keep
moving right 1 and down 3 until you have enough points that you can connect in a line.
Q(x) = 2x2 + 7x – 22
For this, pick different values of x, then plug them into the equation. Once you get an answer (that will be your
y-value), you can plot that point on the graph. When you find enough points, you’ll be able to connect them
into a parabola.
For example, look at x = 3:
Q(3) = 2(3)2 + 7(3) – 22
Q(3) = 2*9 + 21 – 22
Q(3) = 18 + 21 – 22
Q(3) = 18 – 1
Q(3) = 17
Therefore, one point on the graph is (3, 17).
If you find the y-values for lots of x’s, you’ll have multiple points to make a graph with:
x
y
-7
27
-6
8
-5
-7
-4
-18
-3
-25
-2
-28
-1
-27
0
-22
1
-13
2
0
3
17
4
38
5
63
6
92
7
125
The graph looks like this:
20
15
10
5
-10 -9
-8
-7
-6
-5
-4
-3
-2
0
-1 0
-5
1
2
3
4
5
6
7
8
9
10
-10
-15
-20
-25
-30
4. Determine the x-intercepts of g(x) = x2 – x – 20
The x-intercepts are the points when y = 0 (i.e. when g(x) = 0). So, we want to solve the following:
0 = x2 – x – 20
To solve a quadratic equation like this, you can always use the quadratic formula
x = -b ± √( b2 – 4*a*c )
2a
where in this case, a = 1, b = -1, and c = -20.
However, if there is an easy way to factor the equation, you can do that too. (Keep in mind that you can’t factor
everything, but the quadratic equation always works, so if you’re not good at factoring, just use the quadratic
equation for everything.)
Looking at 0 = x2 – x – 20, I want to think of two numbers that we can multiply together to get -20 and that we
can add together to get -1. Two such numbers are -5 and 4. This means that we can factor the equation as...
0 = x2 – x – 20
0 = (x – 5)(x + 4)
(You should multiply this out to confirm that it equals 0 = x2 – x – 20)
When two things multiply together to equal 0, that means that the first one equals 0, the second one equals 0, or
both of them equal 0. In this case, either x – 5 = 0 or x + 4 = 0. Solve both equations:
x–5=0
x=5
x+4=0
x = -4
The equation g(x) = x2 – x – 20 equals 0 when x = 5 and when x = -4 (you can use the quadratic formula to get
the same results). Therefore, the x-intercepts are (-4, 0) and (5, 0).
5. Determine the solution to the following system of equations:
2x – y – 4z = 9
x – 2y + z = 3
3x + y + z = -2
Basically, we want to add the equations together in such a way that variables cancel out, and we are left with a
solution for x, y, or z. Then, we can plug that solution into one or more of the equations, and repeat the process
until we get a solution for all of the variables.
First, multiply the second equation by -2 (if you multiply every term in an equation by one number, making sure
that you multiply everything on both sides of the “=”, this doesn’t change any of the properties of the equation –
i.e. it’s still the same equation).
-2x + 4y - 2z = -6
Now, add that equation to the first one:
-2x + 4y - 2z = -6
+ 2x – y – 4z = 9
0x + 3y – 6z = 3
You get 3y – 6z = 3 (which is the same as y – 2z = 1, dividing everything by 3).
Now, let’s do the same thing with the second and third equations. Multiply the second equation by -3, and add
it to the third equation:
-3x + 6y – 3z = -9
+ 3x + y + z = -2
0x + 7y -2z = -11
This leaves us with two new equations to focus on: y – 2z = 1 and 7y -2z = -11. If you solve the first of these
equations for y, you get y = 2z + 1. Plug that into the second of these equations:
7(2z + 1) – 2z = -11
14z + 7 – 2z = -11
12z + 7 = -11
12z = -18
z = -18/12 = -3/2
Now we have a value for z!
Choose any two of the original equations and plug in our value for z:
2x – y – 4(-3/2) = 9
x – 2y + (-3/2) = 3


2x – y + 6 = 9 
x – 2y – 3/2 = 3 
2x – y = 3
x – 2y = 9/2
So, now we have two new equations, both in terms of x and y. Solve the second of these for x (to get x = 2y +
9/2) and plug that into the other:
2(2y + 9/2) – y = 3
4y + 9 – y = 3
3y + 9 = 3
3y = -6
y = -2
Now, you can plug this into any of the previous equations to get a value for x. However, since we just solved
for x, it’s easiest to just use this equation:
x = 2y + 9/2
x = 2(-2) + 9/2
x = -4 + 9/2
x = -8/2 + 9/2
x=½
In all, we have x = ½, y = -2, and z = -3/2.
Note: If you have 3 unknown variables, you need at least 3 equations to find a solution for each variable. If you
have 2 unknown variables, you only need at least 2 solutions. This is why after we found the value of z, we
only needed to look at 2 of the original equations, because at that point there were only 2 unknown variables – x
and y. Look at multiple choice question 16 for an example of a system of equations with 2 variables.
6. The base of a ladder sits 15 feet from the edge of a building. If the ladder reaches 20 feet up the
building, how long is the ladder?
Whenever you see a word problem like this, the first step is to draw a picture:
--|
|
|
20 ft
|
|
|
|
|
--|--------------|
15 ft
Based on the picture, we have a right triangle with a base of 15 ft and a height of 20 feet. We want to find the
length of the hypotenuse (the length of the ladder).
Remember the Pythagorean Theorem: a2 + b2 = c2, where a and b are the lengths of the base and height, and c is
the length of the hypotenuse. Plug in our values for a and b, and solve for c:
(15)2 + (20)2 = c2
225 + 400 = c2
625 = c2
c = ±√625 = ±25
When you solve for c, you get 25 or -25. However, a ladder can’t be -25 feet long, so the answer must be 25.
The ladder is 25 feet long.
7. Given the formula from physics, v =
v1 + v2 , solve for v1.
1 + v2/c2
In this case, you just have to work systematically until you get v1 alone on one side of the equation:
v=
v1 + v2
1 + v2/c2
Multiply both sides by (1 + v2/c2):
v(1 + v2/c2) = v1 + v2
Subtract v2 from both sides:
v(1 + v2/c2) - v2 = v1
Therefore, v1 = v(1 + v2/c2) - v2.
1. You want 88 ounces of the alloy. Of that 88 ounces, 20.68 ounces will be copper (We know that 23.5% of
the alloy is copper, and 0.235*88 = 20.68).
Call the number of total ounces of the 10% alloy x, and call the number of total ounces of the 28% alloy y. We
know that x + y = 88 (the total ounces of one plus the total ounces of the other = 88 total ounces.
We also know that 0.10x + 0.28y = 20.68 (the ounces of copper in the first alloy plus the ounces of copper in
the second alloy = the ounces of copper in the final alloy).
So, now we have two equations:
x + y = 88
0.10x + 0.28y = 20.68
We can solve these equations like we did in number 5. Solve the first one for y, then plug that result into the
second equation:
y = 88 – x
Plugging that into the second equation:
0.10x + 0.28(88 – x) = 20.68
0.10x + 24.64 – 0.28x = 20.68
-0.18x + 24.64 = 20.68
-0.18x = -3.96
x = 22
Since we only need to know the amount of the 10% alloy, and that value is x, we’re done.
The answer is A.
2. P is the percentage recycled (the y-value of the equation) and t is the years since 1990 (the x value). Based
on the question, we have two points:
In 1995, 25% was recycled  the point is (5, 25)
In 1990, 17% was recycled  the point is (0, 17)
Notice that the y-intercept is at (0, 17). We can solve this problem exactly like question 2.
First, calculate the slope:
25 – 17 = 8/5
5–0
So, we have a slope of 8/5 and a y-intercept of 17. Plug that into the slope-intercept equation of a line:
y = (8/5)x + 17
In the P(t) and t terms of this problem, the equation is:
P(t) = (8/5)t + 17
For the year 2003, t = 13 (2003 – 1990 = 13). Plug that into the equation to get the percentage in the year 2003:
P(13) = (8/5)(13)+ 17
P(13) = 20.8 + 17
P(13) = 37.8
The answer is B.
3. You could multiply out all the answers to see
if any of them equal 15x2 + 19x + 6.
Alternatively, you could try factoring it yourself
and see if your answer is one of the choices.
To factor this, we need to find numbers to go in
the parentheses that will work. The first two
numbers have to multiply together to make 15,
the last two numbers have to multiply together
to get 6, and the sum of the outer numbers
multiplied and the inner numbers multiplied
have to equal 19.
(ax + b)(cx + d)
a*b = 15
a*d + b*c = 19
b*d = 6
Basically, you have to just use trial and error to find numbers that work. The numbers that work in this case
are:
(3x + 2)(5x + 3)
If you multiply this out, you get:
3x*5x + 3x*3 + 2*5x + 2*3
15x2 + 9x + 10x + 6
15x2 + 19x + 6
The answer is D.
4. Same thing for this problem. The first numbers in each set of parentheses have to be 1 (so that if you
multiplied everything out, you’d get an x2 term). Now we need to find numbers that multiply to -35, and add
together to get -1
There is no set of numbers that will work. Look at the numbers that you could multiply together to get -35:
-35
-1
-5
-7
1
35
7
5
None of the pairs of numbers adds together to get -1.
The answer is E.
5. This is a special case. If you look at the first term (49x2), you’ll see that it’s equal to (7x)2. The second term
(4) is equal to (2)2. When you have such a “difference of squares” (something of the form (ax)2 – (b)2), you can
factor it as (ax + b)(ax – b).
This problem can be factored as (7x + 2)(7x – 2).
The answer is A.
Note: Always look for a difference of squares when you’re asked to factor something with only 2 terms (i.e.
with an x2 term and a number, but no x term).
6. You can re-write the original equation by combining the x3 terms, then noticing that this equation looks like
a regular quadratic equation, only with an x3 in place of x:
8x6 + 20x3 + 6x3 + 15 = 8x6 + 26x3 + 15 = 8(x3)2 + 26(x3) + 15 [Looks like 8x2 + 26x + 15]
Answer C works:
(4x3 + 3)(2x3 + 5)
8x6 + 20x3 + 6x3 + 15
8x6 + 26x3 + 15
The answer is C.
7. Since the total sales are $1058.30, this means that the cost of the items (call that x) plus the tax on the items
(at 4%, that would be 0.04x) equals 1058.3. You can write this as:
x + 0.04x = 1058.30
First solve for x:
x + 0.04x = 1058.30
1.04x = 1058.3
x = 1017.596
This is the cost of the items without tax. The tax is 4% of that:
(0.04)(1017.596) = 40.7
The tax was $40.70.
The answer is C.
8. Draw a picture:
10 miles per hour
14 miles per hour 
The two boats are moving apart at a rate of 24 miles per hour (you can add their speeds together).
At this speed, how many hours will it take them to be 48 hours apart? Call the number of hours x, and set up an
equation:
24 miles * x hours = 48 miles
hour
24x = 48
x=2
In 2 hours, they will be 48 miles apart.
The answer is D.
Note: when dealing with time, I like to make sure that all the units of measurement cancel out correctly. Notice
that “miles per hour” translates to miles/hour. In the equation above, the hours cancel out on the left hand side,
leaving only miles, giving us units of miles on both sides of the equation.
9. This is similar to the last problem. The boat is moving at 11 km/hr, and is carried along by the current at 8
km/hr. So, the boat is really moving at 11 + 8 = 19 km/hr. Again, call the number of hours x, and set up an
equation:
19 km/hr * x hrs = 76 km
19x = 76
x = 76/19 = 4
The answer is C.
10. S(x) = 6000 – 600x + 120x2 – 6x3, so S(x) – 6000 = -600x + 120x2 – 6x3
Notice that everything on the right is divisible by 6, so we can factor that out:
S(x) – 6000 = -600x + 120x2 – 6x3
S(x) – 6000 = 6(-100x + 20x2 – x3)
We can also factor out an x:
S(x) – 6000 = 6x(-100 + 20x – x2)
Also, factor out a -1 to make the x2 term positive:
S(x) – 6000 = -6x(100 - 20x + x2)
Now, let’s focus on the terms that are remaining in the parentheses:
x2 - 20 + 100
This can be factored as x2 - 20 + 100 = (x – 10)(x – 10) = (x – 10)2. Plug that into the equation.
S(x) – 6000 = -6x(x – 10)2
The answer is A.
11. First get the absolute value term (|b – 8|) alone on one side of the inequality, then solve for b as in question
1:
|b – 8| - 7 > 8
|b – 8| > 15
b – 8 > 15
b > 23
OR
b – 8 < -15
b < -7
This can be written as (-∞, -7) U (23, ∞), meaning x can be in the interval from negative infinity to -7 or in the
interval from 23 to infinity.
The answer is D.
12. |4m + 2| = 4
4m + 2 = 4
4m = 2
m=½
OR
4m + 2 = -4
4m = -6
m = -3/2
The answer is B.
13. 20y2 + 27y + 9 = 0
Remember that you can always use the quadratic formula. But, this can be factored as (4y + 3)(5y + 3).
Therefore,
4y + 3 = 0
4y = -3
y = -3/4
OR
5y + 3 = 0
5y = -3
y = -3/5
The answer is B.
14. For these problems, make sure that you multiply each term in the first set of parentheses with each term in
the second set of parentheses. It doesn’t matter which order you multiply in, as long as you’re systematic, so
you don’t miss anything.
(3y – 4)(9y2 + 12y + 16)
3y*9y2 + 3y*12y + 3y*16 – 4*9y2 – 4*12y – 4*16
27y3 + 36y2 + 48y – 36y2 – 48y – 64
27y3 + 36y2 + 48y – 36y2 – 28y – 64
27y3 – 64
The answer is B.
15. (x + 2y)(x – 5y)
(x + 2y)(x – 5y)
x*x + x*(-5y) +2y*x + 2y*(-5y)
x2 -5xy + 2xy -10y2
x2 - 3xy - 10y2
The answer is E.
16.
8x – 7y = -48
6x + 4y = -36
When you only have two equations, you can always solve for one variable in the first equation, then plug your
result into the second equation. This is what we’ll do here, and you can follow these steps for any other system
of two equations.
Solve the first equation for x:
8x – 7y = -48
8x = 7y – 48
x = (7/8)y – 6
Plug this in for x in the second equation, then solve for y:
6x + 4y = -36
6((7/8)y – 6) + 4y = -36
(21/4)y – 36 + 4y = -36
(21/4)y + 4y = 0
y=0
Plug the value for y into the equation that we solved for x:
x = (7/8)y – 6
x = (7/8)(0) – 6
x = -6
So, we have x = -6 and y = 0. This corresponds to the point (-6, 0).
The answer is A.
17. 2(y + 2) = 3(y – 6)
When you have one equation with one variable, you need to get all the terms with the variable on one side of
the equation and all the number terms on the opposite side of the equation.
2(y + 2) = 3(y – 6)
2y + 4 = 3y – 18
4 = y – 18
22 = y
The answer is A.
18. 78 = 9x – 3
78 = 9x – 3
81 = 9x
9=x
The answer is C.
19. (1/4)p – (3/8)p = 2
When there are fractions, it’s easiest to multiply the entire equation by a number that will make the
denominators go away. In this case, let’s multiply both sides of the equation by 8:
(1/4)p – (3/8)p = 2
(8/4)p – (24/8)p = 16
2p – 3p = 16
-p = 16
p = -16
The answer is A.
20. Let’s call the number of days d and the number of miles m. We can set up an equation for each of the
customers and solve the system of equations:
108 = 3d + 300m
191 = 5d + 600m
Refer to question 16 about how to solve a system of two equations.
Solve the first equation for d:
108 = 3d + 300m
108 – 300m = 3d
36 – 100m = d
Plug that into the second equation:
191 = 5d + 600m
191 = 5(36 – 100m) + 600m
191 = 180 – 500m + 600m
191 = 180 + 100m
11 = 100m
m = 0.11
Plug that into the equation we solved for d:
36 – 100m = d
36 – 100(0.11) = d
36 – 11 = d
25 = d
The fee per mile is $0.11, or 11 cents. The fee per day is $25
The answer is C.
21. -2x + 7(2x – 6) = -24 – 6x
-2x + 7(2x – 6) = -24 – 6x
-2x + 14x – 42 = -24 – 6x
12x – 42 = -24 – 6x
18x – 42 = -24
18x = 18
x=1
The answer is D.
22. f(x) = |-9x + 2|
The domain of a function is the set of x-values that you can plug into the function (the range is the set of yvalues that the function gives you).
In questions about domain, if the function is a polynomial (something with x’s, x2’s, x3’s, etc., but no x’s in the
denominator of a fraction, and no fractional exponents or square roots) or is the absolute value of a polynomial
(like in this case), the domain is all real numbers.
Remember that you can’t divide anything by 0, so if there is a fraction with an x (or x2’s, x3’s, etc.) term in the
denominator, whatever values of x make the denominator equal to 0 are not in the domain of the function.
For example, look at the function g(x) = 1/(x + 3). The denominator is equal to 0 when x = -3.
Therefore, the domain of this function is all real numbers except x = -3.
Also, remember than you can’t take the square root of a negative number, so if there is an x term in a square
root, whatever values of x make the inside of the square root less than 0 (equal to 0 is ok), will not be in the
domain.
For example, look at the function g(x) = √(x2 – 4). x2 – 4 is less than 0 when x is between -2 and 2. So
the domain is all real number except those between -2 and 2 (this can be written (-∞, -2) U(2, ∞)).
Notice that the values x = -2 and x = 2 are still in the domain.
The answer is A.
23. We have the equation I = Prt, and the values t = 6, P = 170, and r = 0.04. Plug those into the equation and
solve for I:
I = Prt
I = (170)(0.04)(6)
I = 40.8
The interest earned was $40.80.
The answer is B.
24. (1, -3) and (-4, 0)
Notice that this time, we don’t have a y-intercept. This is no problem.
First, solve for the slope:
-3 – 0 = -3
1 –(-4) 5
The slope is -3/5. Plug that into the equation of a line:
y = mx + b
y = (-3/5)x + b
We still need to find b, the y-intercept. We know that both of the points given are on the line (because the
question tells us to find a line containing both points, so they’re on the line by definition). That means that you
can pick either one of the points (I like (-4, 0) because using 0 always makes calculations easier) and plug it into
the equation. So, plug in x = -4 and y = 0, and solve for b:
y = (-3/5)x + b
(0) = (-3/5)(-4) + b
0 = 12/5 + b
-12/5 = b
Now we have the y-intercept (it’s -12/5). Plug that into the equation and we’re done:
y = (-3/5)x – 12/5
The answer is C.
25. f(x) = -x2 + 2x - 5
One way to solve this problem would be to graph it yourself by plotting points. But, let’s see if we can learn
what the graph looks like from the equation, and use that to pick one of the 4 choices.
First of all, look at the coefficient of the x2 term. It’s -1. Any parabola with a negative x2 term will open down
(if the x2 term were positive, it would open up, like a U). Therefore, the possible answers are B and C.
Now, let’s see what the y-intercept is. The y-intercept is where x = 0, so set x = 0 and solve for y:
y = -x2 + 2x – 5
y = -(0)2 + 2(0) – 5
y=0+0–5
y = -5
The y-intercept is at (0, -5). This leaves us only with answer C.
But to be sure, let’s check the x-intercepts. These can be found by setting y = 0 and solving the equation 0 = -x2
+ 2x – 5. We can use the quadratic formula for this:
x = -b ± √( b2 – 4*a*c )
2a
x = -(2) ± √( (2)2 – 4*(-1)*(-5) )
2(-1)
x = -2 ± √(4 – 20)
-2
x = -2 ± √(-16)
-2
The number under the square root is negative, and you can’t find the square root of a negative number, so we
can’t solve for x. This means that there are NO x-intercepts, which is what we would expect given the graph of
C.
The answer is C.