Now comes the nitty gritty of mathematics. Understanding algebraic equations. The ATI TEAS will test the applicant’s ability to simplify equations and solve operations with unknown quantities.
HOW TO SIMPLIFY EQUATIONS
One of the easiest ways to solve algebraic equations is to combine like terms. Like terms is described as terms that contain the same variables (or no variables at all) raised to the same power.
For example: (x2 + 4x +1) + (2×2 + x + 8)
In order to simplify the expression, we must combine like terms. For example, x2 and 2x2 are like terms. They both contain the same variable x raised to the second power (x2). In addition, 4x and x are also like terms because neither has an exponent. Lastly, 1 and 8 are like terms because neither contains a variable. We can combine each set of like terms.
FOIL expressions require the applicant to multiply two binomials. Binomials are defined as equations that contain two same terms. FOIL is primarily used with multiplication of these binomials.
For example: (x + 2) (2x + 4)
FOIL stands for First, Outer, Inner, and Last. This is the order in which we multiple the binominals.
First, we multiple the first two terms of each binomial.
Multiply x by 2x = 2×2
Next, multiply the outer two terms of each binomial.
Multiply x by 4 = 4x
Next, multiply the inner two terms of each binomial.
Multiply 2 by 2x = 4x
Next, multiply the last two terms of each binomial.
Multiply 2 by 4 = 8
Lastly add the results together.
2×2 + 4x + 4x + 8 = 2×2 + 8x + 8
HOW TO SOLVE VARIABLE EQUATIONS
Algebraic equations can become confusing once they involve one unknown variable. This variable is usually represented by the letter x or y. These equations test the applicant’s ability to find the unknown variable.
For example: 3x + 6 = 9
In order to solve these equations, we must isolate the variable on side of the equation. We begin by subtract 6 from both sides of the equal sign. *It’s important to note that what is done on one side of the equal sign is also completed on the other side of the equal sign.
3x + 6 = 9
3x + 6 – 6 = 9 – 6
3x + 0 = 3
3x = 3
Now we divide both sides by 3 to isolate the variable.
3x = 3
x = 1
The value for x is 1.
HOW TO SOLVE INEQUALITY EQUATIONS
Inequalities equations express the relationship between two quantities where one quantity may be greater or less than the other.
Examples of Inequality Symbols
> greater than
< less than
greater than or equal to
less than or equal to
Inequality equations are solved the same way as equations with the exception of one thing. When multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
For example: -3x > 9
In order to solve this equation, we must isolate the variable x on one side of the equation.
3x > 9
** notice the sign change
x < -3
UNDERSTANDING ABSOLUTE VALUE EQUATIONS
The absolute value of a number is the distance that number lies from zero on a number line. Absolute value equations are indicated by two vertical bars:
For example: |X| = 6
If the absolute value of x is equal to 6, then x must lie exactly 6 units away from zero on the number line. This means that x can be either positive (6) or negative (-6). Both 6 and -6 lie exactly 6 units away from zero. Here is another equation example:
|X – 3| = 6
This equation is slightly different. We understand that the absolute value of X – 3 is 6. This informs us that the quantity X – 3 lies equally 6 units away from zero. Meaning, X – 3 could equal 6 or -6. We will set up our equations to solve both possibilities.
X – 3 = 6
X – 3 + 3 = 6 +3
X = 9
X – 3 = – 6
X – 3 + 3 = –6 + 3
X = – 3
The value for X is either 9 or – 3. We set this values in notations, such as {9, –3}.
ATI TEAS Math Algebra
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Question 1 of 8
1. Question
12 – x/4 = 8
Solve the equation above. Which of the following is correct?
Correct
Isolate the variable x on one side of the equation. First, subtract the number 12 from both sides:
12 – x/4 = 8
– x/4 = 8 – 12
– x/4 = -4
Then, multiply both sides by -4 to isolate the variable x:
– x/4 = -4
– x/4 x (- x/4) = -4 x -4
X = 16
The value of x is 16.
Incorrect
Isolate the variable x on one side of the equation. First, subtract the number 12 from both sides:
12 – x/4 = 8
– x/4 = 8 – 12
– x/4 = -4
Then, multiply both sides by -4 to isolate the variable x:
– x/4 = -4
– x/4 x (- x/4) = -4 x -4
X = 16
The value of x is 16.
Question 2 of 8
2. Question
(3x + 2)(x + 1)
Simplify the expression above. Which of the following is correct?
Correct
Use the process of FOIL to multiply the binomials (3x + 2)(x + 1). First, multiply the first two terms of each binomial:
First: 3x x x = 3x2
Then, multiply the outer two terms:
Outer: 3x x 1 = 3x
Then, multiply the inner two terms:
Inner: 2 x x = 2x
Then multiply the last two terms:
Last: 2 x 1 = 2
This expression can be further simplified to 3x2 + 5x + 2.
Incorrect
Use the process of FOIL to multiply the binomials (3x + 2)(x + 1). First, multiply the first two terms of each binomial:
First: 3x x x = 3x2
Then, multiply the outer two terms:
Outer: 3x x 1 = 3x
Then, multiply the inner two terms:
Inner: 2 x x = 2x
Then multiply the last two terms:
Last: 2 x 1 = 2
This expression can be further simplified to 3x2 + 5x + 2.
Question 3 of 8
3. Question
|x + 2| > 4
Solve the inequality above. Which of the following is correct?
Correct
In this inequality, the absolute value of x + 2 is greater than 4. This tells us that the quantity x + 2 lies more than 4 units away from zero on the number line. So, the value of x + 2 could be greater than 4, or it could be less than -4. Set up two inequalities and solve for both possibilities:
X + 2 > 4 x + 2 < -4
X + 2 -2 > 4 – 2 x + 2 – 2 < -4 – 2
X > 2 x < -6
The solution is x > 2 or x < -6.
Incorrect
In this inequality, the absolute value of x + 2 is greater than 4. This tells us that the quantity x + 2 lies more than 4 units away from zero on the number line. So, the value of x + 2 could be greater than 4, or it could be less than -4. Set up two inequalities and solve for both possibilities:
X + 2 > 4 x + 2 < -4
X + 2 -2 > 4 – 2 x + 2 – 2 < -4 – 2
X > 2 x < -6
The solution is x > 2 or x < -6.
Question 4 of 8
4. Question
Cassandra’s height, x, is 3 inches greater than twice her brother’s height, y.
Which of the following algebraic equations best represents the statement above?
Correct
Start with Cassandra’s height, x. Then set up an equation:
X = ?
Cassandra’s brother’s height is denoted by y. Cassandra’s height is equal to 3 more than twice y, which can be written as 2y + 3. Add this expression to the equation:
X = 2y + 3
Incorrect
Start with Cassandra’s height, x. Then set up an equation:
X = ?
Cassandra’s brother’s height is denoted by y. Cassandra’s height is equal to 3 more than twice y, which can be written as 2y + 3. Add this expression to the equation:
X = 2y + 3
Question 5 of 8
5. Question
4(x +7) = 2(x + 15)
Solving the equation above. Which of the following is correct?
Correct
Isolate the variable x on one side of the equation.
First, perform the multiplication on both sides of the equation:
4(x + 7) = 2(x + 15)
4x + 28 = 2x + 30
Then, subtract 28 from both sides:
4x + 28 = 2x + 30
4x + 28 – 28 = 2x + 30 – 28
4x = 2x + 2
Next, subtract 2x from both sides:
4x = 2x + 2
4x – 2x = 2
2x = 2
Now, divide both sides by 2 to isolate the variable x:
2x = 2
2x / 2 = 2 / 2
X = 1
The value of x is 1.
Incorrect
Isolate the variable x on one side of the equation.
First, perform the multiplication on both sides of the equation:
4(x + 7) = 2(x + 15)
4x + 28 = 2x + 30
Then, subtract 28 from both sides:
4x + 28 = 2x + 30
4x + 28 – 28 = 2x + 30 – 28
4x = 2x + 2
Next, subtract 2x from both sides:
4x = 2x + 2
4x – 2x = 2
2x = 2
Now, divide both sides by 2 to isolate the variable x:
2x = 2
2x / 2 = 2 / 2
X = 1
The value of x is 1.
Question 6 of 8
6. Question
(2x2 + 4x -7) – (2x2 + 3x -4)
Simplify the expression above. Which of the following is correct?
Correct
To simplify this expression, combine like terms:
(2x2 + 4x – 7) – (2xx + 3x -4)
= 2x2 – 2x2 + 4x – 3x – 7 – (-4)
= (2x2 – 2x2) + (4x – 3x) – (7 + 4)
= 0 + x – 3
= x – 3
The simplified expression is x – 3.
Incorrect
To simplify this expression, combine like terms:
(2x2 + 4x – 7) – (2xx + 3x -4)
= 2x2 – 2x2 + 4x – 3x – 7 – (-4)
= (2x2 – 2x2) + (4x – 3x) – (7 + 4)
= 0 + x – 3
= x – 3
The simplified expression is x – 3.
Question 7 of 8
7. Question
The value of x is less than 3/4 the value of y.
Which of the following algebraic expressions correctly represents the sentence above?
Correct
Start with the value if x, and set up an equation:
X = ?
The question tells us that x is 5 less than 3/4 the value of y, which can be written as 3/4 y – 5. Add this expression into the equation:
X =3/4 y – 5
Incorrect
Start with the value if x, and set up an equation:
X = ?
The question tells us that x is 5 less than 3/4 the value of y, which can be written as 3/4 y – 5. Add this expression into the equation:
X =3/4 y – 5
Question 8 of 8
8. Question
|9 – x| = 4
Which of the following is the solution set for the equation above?
Correct
In this equation, the absolute value of 9 – x is 4. This tells us that the quantity 9 – x lies exactly 4 units away from zero on the number line. So, 9 – x could equal 4 or -4. Set up two equations and solve for both possibilities:
9 – x = 4 9 – x = -4
9 – 9 – x = 4 – 9 9 – 9 – x = -4 – 9
-x = -5 -x = -13
x = 5 x = 13
The value of x is 5 or 13. In set notation, this is written as {5, 13}.
Incorrect
In this equation, the absolute value of 9 – x is 4. This tells us that the quantity 9 – x lies exactly 4 units away from zero on the number line. So, 9 – x could equal 4 or -4. Set up two equations and solve for both possibilities:
9 – x = 4 9 – x = -4
9 – 9 – x = 4 – 9 9 – 9 – x = -4 – 9
-x = -5 -x = -13
x = 5 x = 13
The value of x is 5 or 13. In set notation, this is written as {5, 13}.
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