ACT Math Practice
Algebra Placement Test
The Algebra Placement Test is composed of items from three curricular areas:
elementary algebra, coordinate geometry, and intermediate algebra. Each of these
three areas is further subdivided into a number of more specific content areas. Overall,
the Algebra Placement Test includes items from more than 20 content areas; however,
the majority of test questions fall within the following eight content areas:
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1. Substituting Values into Algebraic Expressions
2. Setting Up Equations for Given Situations
3. Basic Operations with Polynomials
4. Factoring Polynomials
5. Linear Equations in One Variable
6. Exponents and Radicals
7. Rational Expressions
8. Linear Equations in Two Variables
Substitute -3 in for all x and evaluate using your
order of operations. (-3)2 -1 = 8 and then (-3)+1
= -2. 8/-2 = -4.
Remember if you are using a calculator,
evaluate the numerator and denominator
separately or be sure to use () around the
whole denominator and the whole numerator.
You must also use () around each of the -3
First, find this person’s MHR by
subtracting 220 – 43 = 177. Then
substitute the MHR and provided RHR
to get
THR = 54+.65(177-54).
THR = 54 +.65(123)
THR = 54+79.95
THR = 133.95
Because you want to do the subtraction first and then
take 75% of the answer, you need to put () around the
220-a to get that value 1st then multiply by .75.
Make sure that when you set up an equation
here, you are using distance on both sides of
the equation. The word “entire” suggests
total and that means addition. First part of
the flight is 8 hours at some speed x , thus 8x.
Second part is 7 hours at 325 mph, thus
7(325). The plane flew an average airspeed
of 350 and that was for 15 hours so….
Follow order of operations and begin by
getting rid of () by distributing the negative
through (-6a-3b) to get 3a+4b+6a+3b. Then
combine like terms: 9a+7b.
Here you want to combine like terms . Like
terms have to the same variables and the
same exponent for the variables. The only
two that are like that here are 2a2b2 and
a2b2. Since they are alike, add their
coefficients, 2 and the understood 1, to get
3a2b2. Leave the order two terms as they
are. The terms can be in any order here.
To factor a trinomial with a one for a
coefficient, you need to note the constant or
number with a variable and read backwards to
the middle term. Here we are looking for
factors of 20 that will subtract (because of the
minus sign) to give you a -1 (coefficient of x
part. Factors of 20 are 1 and 20, 2 and 10, and
5 and 4. Five and four will subtract to give you
a one. In order to get a -1, the five has to be
negative so: x-5 works.
To factor a trinomial with a one
for a coefficient, you need to note
the constant or number with a
variable and read backwards to
the middle term. Here we are
looking for factors of 6 that will
subtract (because of the minus
sign) to give you a -5 (coefficient
of x part. Factors of 6 are 1 and 6,
2 and 3, Six and one will subtract
to give you a five. In order to get
a -5, the six has to be negative so:
x-6 works.
To solve equations your goal is to get the
variable by itself on one side of the equation.
So here, you need to first distribute and then
combine any like terms on one side of the
equation. Here distribute the 2 through the (),
yielding 2x-10 = -11. There are no terms on
each side that can be combined so we then add
+10 on both sides to get the 2x by itself. Then
gives you 2x = -1. Divide both sides by 2 and
get x = -1/2.
Remember you can always check an equation
by substituting the answer back it to see if it
works.
I like to multiply everything through by the
least common denominator here to get rid of
all of the fractions. This number is 10. If I
multiply EVERY term by 10, I get:
8+-3 = 10x+15. Next, combine like terms on
the left to get 5 = 10x+15. Then, subtract 15
from both sides and get -10 = 10x. Divide
both sides by 10 and get -1.
Method 2: Write it all out:
4*4*r*r*r*t*z*z*z*z*z
-4*r*t*t*t*z*z
Then mark out everything in common and
your answer is what is left.
Method 1 - In order to reduce a monomial
algebraic fraction, begin by reducing the
number portion like you always have: 16/4 leaves -4 in the numerator. Then, you do
the variable portion. Since we are dividing,
up steps up says we need to SUBTRACT
exponents and keep the answer where the
larger number of variables to begin with.
For the r’s, 3-1 is 2, leaving an r2 in the
numerator. Follow the same process for
the t’s, and there will be a t2 left in
denominator. There will be a z3 left in the
numerator for the z’s so…
It is incorrect to leave a radical in the
denominator, so we have to deal with this. The
way to get rid of a radical in the denominator is
to multiply both top and bottom by the
CONJUGATE of the denominator. The conjugate
is the same terms in the denominator both with
the sign changed in the middle. Ours is 3√x + √y.
So √x (3√x + √y) / (3√x - √y)(3√x + √y). Simplify
by distributing and FOILING. The numerator
becomes 3x + √xy because the x*x gets rid of the
radical in the first term. The denominator
becomes 9x – y because the x*x and the y*y
gets rid of the radical and the middle terms from
FOIL add up to 0 or cancel each other.
In order to reduce a multi-term algebraic fraction,
you must factor everything first. Here only the
numerator factors. Using the concepts from before,
the factors of 32 that will add to be 12 are 8 and 4,
and they must be positive to add to positive. So, we
have: (x+8)(x+4) / (x+4) . Because the binomial x+4
terms are exactly alike, they reduce to 1 and can be
marked out, leaving x+8.
Hint: since none of the answers have a denominator
(x+4) had to be one of factors of the numerator.
Again, we need to factor first. The numerator is a
difference of squares so it factors to (3-x)(3+x). So we
have (3-x)(3+x) / (x-3). 3-x and x-3 are exact
opposites so we can rewrite the denominator to -1(3x). Then the 3-x from the top and bottom will reduce
leaving (3+x) / -1. Divide the numerator by the -1 and
get -3-x which is the same as –x-3. BE CAREFUL WITH
SIGNS HERE!!!
The most common process taught is to solve for
y and the slope is the your coefficient of x. So,
here we subtract the 6 and the 2x over to the
other side leaving 3y = -2x -6. Then divide by 3
to get the y isolated. This leaves y = -2/3 x+ -2.
The slope is -2/3 because that is the coefficient
of x.
You can also graph the line using intercepts and
count the slope, knowing that slope is rise over
run.
Plot A and graph the line x =2. X =
2 is vertical crossing the x-axis at 2.
A is 6 units from the vertical line so
if x=2 is going to bisect segment
AB, then the other endpoint must
also be 6 units on the other side of
the line on a horizontal line so….
Remember perpendicular lines
have to intersect at right angles.
Horizontal and vertical lines make it
easier because their slopes are
special cases.
A.
6
6