How to Use Opposite Operations
When solving an equation in algebra, you will often use an opposite operation such as additive inverse or multiplicative inverse (reciprocal) to work your way toward the answer. You have to undo operations that have been done to the variable. The opposite of an operation is another operation that gets you back where you started. This is used primarily to get rid of numbers that are combined with a variable so you can solve for the variable in an equation.
When you need to use the opposite of basic operations — addition, subtraction, multiplication, and division — you need to remember how the additive inverse and multiplicative inverse work:
The additive inverse is the number with the opposite sign. So −3 is the additive inverse of 3, and 16 is the additive inverse of −16. Use these if 3 or 16 is being added to a variable, and you want to get the variable alone; this is used when solving an equation for the value of the variable.
For example, if you have the problem 4 + x = 10, you have to use an opposite operation to solve for the variable x.
4 + x = 10
(4 – 4) + x = (10 – 4)
0 + x = 6
x = 6
The multiplicative inverse is also called the reciprocal. The reciprocal is the original number written as the bottom of a fraction with a 1 on the top. So 1/2 is the reciprocal of 2. If a number starts out as a fraction, its reciprocal is just that number written upside-down. So the reciprocal of 4/7 is 7/4. Similarly, the reciprocal of 1/25 is 25/1, or simply 25.
You use a reciprocal if a number multiplies or divides a variable; it gets the variable alone so it can be solved for. For example, if you have the problem 5x = 20, you have to use an opposite operation to solve for the variable x.
In algebra, you use more than just the basic mathematical operations, so when you need to solve an algebra problem using opposite operations, remember this list:
The opposite of addition is subtraction.
The opposite of subtraction is addition.
The opposite of multiplication is division.
The opposite of division is multiplication.
The opposite of taking a square root is squaring (multiplying a value by itself).
The opposite of squaring is taking the square root.
The opposite of cubing is taking the cube root.
Algebra Glossary
absolute value
An operation that tells you how far a number is from zero.
Algebra Glossary
additive inverse
The number with the same numerical part but the opposite sign (plus or minus) of a given number. If zero is the sum of two numbers, then these two numbers are additive inverses of one another.
Algebra Glossary
associative property
A characteristic of addition and multiplication that allows the grouping of terms to change without affecting the result.
Algebra Glossary
base
A value multiplied repeatedly in an exponential expression.
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binary operation
A process requiring two values to produce a third value.
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binomial
Two terms separated by addition or subtraction.
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braces
A grouping symbol that looks like this: { }.
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brackets
A grouping symbol that looks like this: [ ].
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coefficient
A number multiplied by a variable. The coefficient precedes the variable.
Algebra Glossary
combinations
A method of counting that tells how many ways a designated number of objects can be selected from a given set.
Algebra Glossary
common denominator
The same value on the bottom of more than one fraction.
Algebra Glossary
commutative property
A characteristic of addition and multiplication that allows the order of the values in an operation to be changed without affecting the result.
Algebra Glossary
composite number
A whole number larger than 1 that isn’t prime.
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constant
A variable or number that never changes in value in an expression.
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cube; perfect cube
The third power of a number; the result of multiplying a number by itself three times.
Algebra Glossary
cube root
A number that you can multiply by itself three times to get a given number. For example, the cube root of 8 is 2, because 2 times 2 times 2 equals 8.
Algebra Glossary
cubic
An adjective that describes an expression in which the highest power is three.
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decimal
A fraction with an unwritten denominator of 10 indicated by the decimal point.
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degree of an expression
The highest power occurring in an expression.
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denominator
The bottom number in a fraction.
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difference
The result of subtraction.
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digit
Numerals from zero through nine, so called because they were originally counted on the fingers.
Algebra Glossary
distributive property
A characteristic of multiplication and addition that allows for the multiplication of each individual term in a grouped series by a term outside of the grouping without changing the value of the expression.
Algebra Glossary
dividend
A number to be divided by another number.
Algebra Glossary
divisible
A number that can be divided by another number with no remainder.
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double root
A solution that appears twice when solving an equation because the related factor appears twice in the factored form. For example, in (y – 2)(y – 2) = 0, y = 2 is a double root or solution. In (y – 2)(y – 2)(x + 3) = 0, y = 2 is still a double root or solution.
Algebra Glossary
equation
A mathematical statement with an equal sign showing that two values are equal.
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equivalent fractions
Fractions equal to one another, even though they may have different denominators.
Algebra Glossary
even number
A number that can be divided by 2.
Algebra Glossary
exponent; power
A value in smaller type found above and to the right of the base that indicates the number of times the base is multiplied by itself.
Algebra Glossary
expression
Any combination of values (variables, numbers, and/or constants) and operations that can be used to show how things belong together and compare to one another.
Algebra Glossary
factor
1. (noun) Any of the values involved in a multiplication problem that when multiplied together produces a result. 2. (verb) To rewrite an algebraic expression as a product.
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factorial
An operation that multiplies a whole number by every counting number smaller than the whole number.
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FOIL
1. An acronym for first, outer, inner, and last — which indicates the order in which two terms (one from each of two binomials) are multiplied together. 2. The process of multiplying two binomials.
Algebra Glossary
fraction
Any quantity expressed as a numerator (the value above the fraction line) and a denominator (the value below the fraction line).
Algebra Glossary
fraction line
A line separating the numerator from the denominator in a fraction.
Algebra Glossary
greatest common factor; GCF
The largest possible value that evenly divides each term of an expression containing two or more terms.
Algebra Glossary
grouping symbol
Parentheses, brackets, and braces that can affect the order of operations. Terms and operations within the grouping symbol take precedence.
Algebra Glossary
groups
Terms and operations within grouping symbols.
Algebra Glossary
improper fraction
A fraction whose numerator is larger than the denominator.
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inequality
A relationship between two unequal values.
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infinite
Without end; uncountable, as in a repeating decimal.
Algebra Glossary
integer
A positive or negative whole number or zero; numbers starting with zero and going up or down in increments of one.
Algebra Glossary
irrational number
A number with no fractional equivalent whose decimal never repeats or terminates.
Algebra Glossary
linear
An adjective describing an expression or equation in which the highest power of any variable is one. Constants can be higher powers.
Algebra Glossary
lowest terms
The status of a fraction whose numerator and denominator cannot be divided evenly by the same number.
Algebra Glossary
mixed number
An improper fraction written as a whole number alongside a fraction.
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monomial
An expression with only one term.
Algebra Glossary
multiple
A number evenly divisible by a specific factor.
Algebra Glossary
multiplication property of zero; MPZ
A rule stating if the product of two numbers is zero, then one of the numbers must be zero.
Algebra Glossary
multiplicative inverse
Also known as a reciprocal, one of two numbers whose product is one. The reciprocal of a number is that particular number in the denominator of a fraction with a value of one in the numerator.
Algebra Glossary
multiplier
An integer multiplied with both the numerator and denominator of a fraction.
Algebra Glossary
natural numbers; counting numbers
Values starting with one and increasing by one.
Algebra Glossary
negative number
Any quantity that is less than zero; it is usually preceded by a minus sign.
Algebra Glossary
negative reciprocals
Two numbers, one positive and one negative, whose product is –1.
Algebra Glossary
nested
A condition where two or more grouping symbols are inside one another.
Algebra Glossary
number system
A set of numbers.
Algebra Glossary
numerator
The top number in a fraction.
Algebra Glossary
odd number
A whole number that is not evenly divided by 2.
Algebra Glossary
operation
A mathematical process, such as addition, subtraction, multiplication, and division, performed on one or more quantities.
Algebra Glossary
percent
Fractions with a denominator of 100. The percentage is the numerator of the fraction, indicating the number of parts out of 100.
Algebra Glossary
permutation
A counting method that determines the number of ordered arrangements there are when a certain number of objects are selected from a given set.
Algebra Glossary
polynomial
An expression with one or more terms.
Algebra Glossary
positive number
Any quantity greater than zero.
Algebra Glossary
power
A value of an exponent indicating the number of times the base is multiplied by itself.
Algebra Glossary
power of zero
A rule of exponents, where any number raised to the power of zero equals one as long as the base number isn’t zero.
Algebra Glossary
prime factorization
The process of finding the prime numbers that, when multiplied together, produce a given composite number.
Algebra Glossary
prime number
A whole number larger than one that can be divided evenly only by itself and one.
Algebra Glossary
principal square root
A positive number that when multiplied by itself produces a given positive number.
Algebra Glossary
product
The result of multiplication.
Algebra Glossary
proper fraction
A fraction whose value is less than one; thus, the numerator is always smaller than the denominator.
Algebra Glossary
proportion
An equation showing that two ratios are equal to one another.
Algebra Glossary
quadratic
Also known as second degree, an expression or equation in which the highest power is 2.
Algebra Glossary
quotient
The result of division.
Algebra Glossary
radical
The symbol for the operation to find a square root.
Algebra Glossary
raised to the power of
Multiplying a base number the number of times indicated by the exponent.
Algebra Glossary
rational number
A quantity, positive or negative, that can be written as a fraction; its decimal equivalent terminates or repeats.
Algebra Glossary
real number
Any rational or irrational number.
Algebra Glossary
reciprocal; multiplicative inverse
1. Two numbers whose product is always one. 2. Either one of the two numbers in a reciprocal.
Algebra Glossary
reduce
To divide out a common factor of the numerator and denominator of a fraction, leaving an equivalent fraction.
Algebra Glossary
relatively prime
The condition of two numbers that have no factors in common other than the number one.
Algebra Glossary
remainder
A value that is left over when one number is divided by another.
Algebra Glossary
repeating decimal; recurring decimal
A decimal in which, beyond a certain point, a digit or set of digits repeats indefinitely.
Algebra Glossary
root
A value that, multiplied by itself a number of times, results in the value or number wanted.
Algebra Glossary
rounding
Approximating a value to the nearest digit or decimal place.
Algebra Glossary
scientific notation
A standard way of writing very large and very small numbers as the product of two values — a number between 1 and 9 and a power of 10. Scientific notation follows the form N × 10a where N is a number from 1 up to 10, but not 10 itself, and a is an integer (positive or negative number).
Algebra Glossary
sign
A symbol indicating whether a value is positive (+) or negative (−).
Algebra Glossary
simple fraction
A fraction in which both the numerator and the denominator are whole numbers.
Algebra Glossary
simplify
To combine all that can be combined in an expression or equation, and put it in its most easily understandable form.
Algebra Glossary
solution of equation
Value(s) of a variable that make an equation a true statement.
Algebra Glossary
solve
Find the answer or the number that a variable stands for.
Algebra Glossary
square; perfect square
1. The product of another number times itself. 2. A value with an exponent of 2.
Algebra Glossary
square root
A value resulting when a given value is multiplied by itself.
Algebra Glossary
substitution
A method of replacing a value with its equivalent.
Algebra Glossary
sum
The result of addition.
Algebra Glossary
symmetric property
A characteristic of equations that allows for the exchange of the value(s) on one side of the equal sign with the value(s) on the other side (quantities on the right go to the left; quantities on the left go to the right) without changing the truth of the equation: If x = y, then y = x.
Algebra Glossary
synthetic division
A short-cut division process in which only the coefficients of the terms in an expression are used. The answer is obtained by multiplying and adding.
Algebra Glossary
term
A group of number(s) and/or variable(s) connected to one another by multiplication or division and separated from other terms by addition or subtraction.
Algebra Glossary
trinomial
An expression with three terms. Each term is separated from the others by addition or subtraction.
Algebra Glossary
unFOIL
To factor a trinomial into two binomials.
Algebra Glossary
value
A numeric equivalent or worth of an expression or variable.
Algebra Glossary
variable
A letter representing an unknown number or what you’re solving for in an algebra problem.
Algebra Glossary
whole numbers
All natural numbers plus zero.
