Term
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Definition
A set is a collection of objects.
Those objects are generally called members or elements of the set. |
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Term
| Natural (or Counting) Numbers |
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Definition
N = {1, 2, 3, 4, 5, ...}
Makes sense, we start counting with the number 1 and continue with 2, 3, 4, 5, and so on. |
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Term
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Definition
{0, 1, 2, 3, 4, 5, ...}
The only difference between this set and the one above is that this set not only contains all the natural numbers, but it also contains 0, where as 0 is not an element of the set of natural numbers. |
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Term
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Definition
Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}
This set adds on the negative counterparts to the already existing whole numbers (which, remember, includes the number 0).
The natural numbers and the whole numbers are both subsets of integers. |
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Term
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Definition
Q = {[a over b] a and b are integers and [b not equal to 0]}
In other words, a rational number is a number that can be written as one integer over another.
Be very careful. Remember that a whole number can be written as one integer over another integer. The integer in the denominator is 1 in that case. For example, 5 can be written as 5/1.
The natural numbers, whole numbers, and integers are all subsets of rational numbers. |
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Term
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Definition
I = {x | x is a real number that is not rational}
In other words, an irrational number is a number that can not be written as one integer over another. It is a non-repeating, non-terminating decimal.
One big example of irrational numbers is roots of numbers that are not perfect roots - for example or . 17 is not a perfect square - the answer is a non-terminating, non-repeating decimal, which CANNOT be written as one integer over another. Similarly, 5 is not a perfect cube. It's answer is also a non-terminating, non-repeating decimal.
Another famous irrational number is (pi). Even though it is more commonly known as 3.14, that is a rounded value for pi. Actually it is 3.1415927... It would keep going and going and going without any real repetition or pattern. In other words, it would be a non terminating, non repeating decimal, which again, can not be written as a rational number, 1 integer over another integer. |
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Term
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Definition
R = {x | x corresponds to point on the number line}
Any number that belongs to either the rational numbers or irrational numbers would be considered a real number. That would include natural numbers, whole numbers and integers.
Above is an illustration of a number line. Zero, on the number line, is called the origin. It separates the negative numbers (located to the left of 0) from the positive numbers (located to the right of 0).
I feel sorry for 0, it does not belong to either group. It is neither a positive or a negative number. |
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Term
Order Property for
Real Numbers |
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Definition
Given any two real numbers a and b,
if a is to the left of b on the number line, then a < b.
If a is to the right of b on the number line, then a > b. |
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Term
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Definition
The absolute value of x, notated |x|, measures the DISTANCE that x is away from the origin (0) on the real number line.
Aha! Distance is always going to be positive (unless it is 0) whether the number you are taking the absolute value of is positive or negative. |
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Term
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Definition
a/b , where b not equal to 0
a = numerator
b = denominator |
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Term
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Definition
When you rewrite a number using prime factorization, you write that number as a product of prime numbers.
For example, the prime factorization of 12 would be
12 = (2)(6) = (2)(2)(3).
A prime number is a whole number that has two distinct factors, 1 and itself.
Examples of prime numbers are 2, 3, 5, 7, 11, and 13. The list can go on and on.
Be careful, 1 is not a prime number because it only has one distinct factor which is 1.
That last product is 12 and is made up of all prime numbers. |
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Term
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Definition
| A fraction is simplified if the numerator and denominator do not have any common factors other than 1. You can divide out common factors by using the Fundamental Principle of Fractions |
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Term
| Fundamental Principle of Fractions |
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Definition
a*c/b*c = a/b
In other words, if you divide out the same factor in both the numerator and the denominator, then you will end up with an equivalent expression. An equivalent expression is one that looks different, but has the same value. |
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Term
Writing the Fraction in Lowest Terms
(or Simplifying the Fraction) |
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Definition
Step 1: Write the numerator and denominator as a product of prime numbers.
Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors. |
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Term
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Definition
a*c/b*d = ac/bd
In other words, when multiplying fractions, multiply the numerators together to get the product’s numerator and multiply the denominators together to get the product’s denominator.
Make sure that you do reduce your answers, |
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Term
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Definition
a/1 * 1/a = 1
Two numbers are reciprocals of each other if their product is 1.
In other words, you flip the number upside down. The numerator becomes the denominator and vice versa.
For example, 5 (which can be written as 5/1) and 1/5 are reciprocals. 3/4 and 4/3 are also reciprocals of each other. |
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Term
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Definition
a/b div by c/d = a/b * d/c = ad/bc
In other words, when dividing fractions, use the definition of division by rewriting it as multiplication of the reciprocal and then proceed with the multiplication. |
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Term
Adding or Subtracting Fractions
with Common Denominators |
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Definition
a/c + b/c = a+b/c
Step 1: Combine the numerators together.
Step 2: Put the sum or difference found in step 1 over the common denominator.
Step 3: Reduce to lowest terms if necessary. |
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Term
| Why do we have to have a common denominator when we add or subtract fractions? |
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Definition
| The denominator indicates what type of fraction that you have and the numerator is counting up how many of that type you have. You can only directly combine fractions that are of the same type (have the same denominator). |
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Term
| Least Common Denominator (LCD) |
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Definition
| The LCD is the smallest number divisible by all the denominators. |
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Term
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Definition
| Equivalent fractions are fractions that look different but have the same value. |
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Term
Rewriting Mixed Numbers as
Improper Fractions |
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Definition
In some problems you may start off with a mixed number and need to rewrite it as an improper fraction. You can do this by multiplying the denominator times the whole number and then add it to the numerator. Then, place this number over the existing denominator.
An improper fraction is a fraction in which the numerator is larger than the denominator. |
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Term
Adding or Subtracting Fractions
Without Common Denominators |
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Definition
Step 1: Find the Least Common Denominator (LCD) for all denominators.
Step 2: Rewrite fractions into equivalent fractions with the common denominator.
Step 3: Add and subtract the fractions with common denominators as described above. |
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Term
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Definition
An exponent tells you how many times that you write a base in a PRODUCT.
In other words, exponents are another way to write MULTIPLICATION. |
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Term
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Definition
Please Parenthesis or grouping symbols
Excuse Exponents (and radicals)
My Dear Multiplication/Division left to right
Aunt Sally Addition/Subtraction left to right |
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Term
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Definition
| A variable is a letter that represents a number. |
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Term
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Definition
| An algebraic expression is a number, variable or combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents, and/or roots. |
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Term
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Definition
| You evaluate an expression by replacing the variable with the given number and performing the indicated operation. |
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Term
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Definition
When you are asked to find the value of an expression, that means you are looking for the result that you get when you evaluate the expression.
So keep in mind that vary means to change - a variable allows an expression to take on different values, depending on the situation.
For example, the area of a rectangle is length times width. Well, not every rectangle is going to have the same length and width, so we can use an algebraic expression with variables to represent the area and then plug in the appropriate numbers to evaluate it. |
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Term
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Definition
| Two expressions set equal to each other. |
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Term
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Definition
A value, such that, when you replace the variable with it,
it makes the equation true.
(the left side comes out equal to the right side) |
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Term
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Definition
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Term
Adding Real Numbers
with the Same Sign |
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Definition
Add the absolute values.
Attach their common sign to sum.
In other words:
If both numbers that you are adding are positive, then you will have a positive answer.
If both numbers that you are adding are negative then you will have a negative answer. |
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Term
Adding Real Numbers
with Opposite Signs |
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Definition
Take the difference of the absolute values.
Attach the sign of the number that has the higher absolute value. |
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Term
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Definition
Opposites are two numbers that are on opposite sides of the origin (0) on the number line, but have the same absolute value. In other words, opposites are the same distance away from the origin, but in opposite directions.
The opposite of x is the number -x.
Keep in mind that the opposite of 0 is 0. |
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Term
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Definition
For every real number a,
-(-a) = a. |
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Term
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Definition
| In other words, to subtract b, you add the opposite of b. |
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Term
Multiplicative Inverse
(or reciprocal) |
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Definition
A more common term used to indicate a multiplicative inverse is the reciprocal.
A multiplicative inverse or reciprocal of a real number a (except 0) is found by “flipping” a upside down. The numerator of a becomes the denominator of the reciprocal of a and the denominator of a becomes the numerator of the reciprocal of a.
When you take the reciprocal, the sign of the original number stays intact. |
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Term
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Definition
If a and b are real numbers and
b is not 0, then
a/b = a * 1/b |
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Term
| Multiplying or Dividing Real Numbers |
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Definition
Step 1: Multiply or divide their absolute values.
Step 2: Put the correct sign.
If the two numbers have the same sign, the product or quotient is positive.
If they have opposite signs, the product or quotient is negative. |
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Term
Multiplying by and
Dividing into Zero |
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Definition
a(0) = 0
and
0/a = 0 (when a does not equal 0) |
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Term
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Definition
| Zero (0) does not go into any number, so whenever you are dividing by zero (0) your answer is undefined. |
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Term
The Commutative Properties of
Addition and Multiplication |
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Definition
| The Commutative Property, in general, states that changing the ORDER of two numbers either being added or multiplied, does NOT change the value of it. |
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Term
The Associative Properties of
Addition and Multiplication |
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Definition
| The Associative property, in general, states that changing the GROUPING of numbers that are either being added or multiplied does NOT change the value of it. Again, the two sides are equivalent to each other. |
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Term
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Definition
| In other words, when you have a term being multiplied times two or more terms that are being added (or subtracted) in a ( ), multiply the outside term times EVERY term on the inside. |
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Term
| when you have a negative sign in front of a ( ) |
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Definition
| you can think of it as taking a -1 times the ( ). What you end up doing in the end is taking the opposite of every term in the ( ) |
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Term
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Definition
The additive identity is 0
a + 0 = 0 + a = a |
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Term
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Definition
Multiplication identity is 1
a(1) = 1(a) = a |
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Term
| Additive Inverse (or negative) |
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Definition
For each real number a, there is a unique real number,
denoted -a, such that
a + (-a) = 0. |
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Term
Multiplicative Inverse
(or reciprocal) |
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Definition
For each real number a, except 0, there is a unique real number 1/a such that
a*1/a = 1/a*a = 1
When you take the reciprocal, the sign of the original number stays intact. Remember that you need a number that when you multiply times the given number you get 1. If you change the sign when you take the reciprocal, you would get a -1, instead of 1, and that is a no no. |
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Term
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Definition
| Examples of terms are 3x, 5yexponent3, 2ab, z |
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Term
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Definition
| A coefficient is the numeric factor of your term. (factor = part) In 3x, 3 is the coefficient. |
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Term
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Definition
A constant term is a term that contains only a number. In other words, there is no variable in a constant term.
Examples of constant terms are 4, 100, and -5. |
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Term
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Definition
Like terms are terms that have the exact same variables raised to the exact same exponents.
One example of like terms is 3xTothe2ndpower and -5xTothe2ndpower. Another example is . |
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Term
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Definition
You can only combine terms that are like terms. You think of it as the reverse of the distributive property.
It is like counting apples and oranges. You just count up how many variables you have the same and write the number in front of the common variable part.
Like 7a - 10a = (7-10)a = -3a |
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Term
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Definition
| Two expressions set equal to each other. |
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Term
Linear Equation
in One Variable |
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Definition
Linear Equation
in One Variable
An equation that can be written in the form
ax + b = c
where a, b, and c are constants. |
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Term
| Linear equation exponent value |
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Definition
| Note that the exponent (definition found in Tutorial 4: Introduction to Variable Expressions and Equations) on the variable of a linear equation is always 1. |
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Term
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Definition
A value, such that, when you replace the variable with it,
it makes the equation true. |
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Term
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Definition
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Term
Solving a Linear Equation
in General |
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Definition
| Get the variable you are solving for alone on one side and everything else on the other side using INVERSE operations. |
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Term
| Addition and Subtraction Properties of Equality |
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Definition
If a = b, then a + c = b + c
If a = b, then a - c = b - c |
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Term
| Note that addition and subtraction are inverse operations of each other. |
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Definition
| Note that addition and subtraction are inverse operations of each other. For example, if you have a number that is being added that you need to move to the other side of the equation, then you would subtract it from both sides of that equation. |
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Term
Linear Equation
in One Variable |
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Definition
An equation that can be written in the form
ax + b = c
where a, b, and c are constants. |
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Term
| Multiplication and Division Properties of Equality |
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Definition
If a = b, then a(c) = b(c)
If a = b, then a/c = b/c where c is not equal to 0. |
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Term
| Note that multiplication and division are inverse operations of each other. |
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Definition
| For example, if you have a number that is being multiplied that you need to move to the other side of the equation, then you would divide it from both sides of that equation. |
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