Math 32: Module III Linear Equations and Inequalities in One Variable
By following certain procedures, we can often transform an equation into a simpler equivalent equation that has the form of x = some number.
In this form, the number is a solution of the equation.
Interval Notation is another way to represent the solution to an inequality.
It uses two values -- the starting point and the end point of the interval representing the solution. These two values are written inside (parentheses) and/or [brackets].
Just as an open circle is used on the number line to represent < or >, a parentheses is used in interval notation.
Likewise, a closed circle is used on the number line to represent ≤ or ≥, a bracket is used in interval notation.
Natural Numbers
Natural numbers are all whole numbers, excluding zero. The set of natural numbers is usually represented by N. The set of natural numbers could be written as N = {1, 2, 3, 4, 5...} Sets are represented by capital letters. The elements in the set are contained in braces and separated by commas. The set of natural numbers in an infinite set. The list of numbers continues forever, and is indicated by three dots, which are also known as an ellipsis.
A = {consonants} B = {letters of the alphabet}
Set A is a subset of B. this can be written as A⊆B.
Set B is the set of natural numbers between -6 and 0.
Set B is an empty set, or a null set. We can write this as { } or Ø.
*NOTE
Since division can be performed by multiplying by the reciprocal, this property works for division as well. a, b, and c with c ≠ 0, if a = b, then a/c = b/c
*NOTE
Since subtraction can be defined in terms of addition, this property works for subtraction as well as addition. If a = b, then a - c = b - c
Solving equations when simplifying is needed
Sometimes it is necessary to simplify one or both sides of the equation before getting the variable term on one side of the equation and the number term on the other side of the equation. Simplify by combining like terms that are on the same side of the equation.
Solving Equations with Fractions
The equation-solving procedures is the same for equations with or without fractions, however, takes care and can be time consuming.
Set Intersection
The intersection of two sets is the set of all the elements that are common to both sets. The intersection of A and B is written as A⋂B and includes those elements that are in set A and in set B. If A = {a, b, c, d, e} and B = {a, e, i, o, u}, then A⋂B = {a, e}
Least Common Denominator (LCD)
The least common denominator (LCD) of two or more fractions is the least common multiple (LCM) of the denominators of the fractions.
When writing the notation for an empty set, only use one of the symbols -- either { } or Ø, but not both.
The notation { Ø } means "a set whose only element is the empty set."
Solving the Equation
The process of finding the solution(s) of an equation is called solving the equation. The goal of solving the equation is to get the variable alone on one side of the equation. x = some number or some number = x
Set Union
The union of two sets is the set of every element that is in either or both sets. The union of sets A and B is written as A⋃B and includes those elements in either set A, set B, or both. If A = {a, b, c, d, e} and B = {a, e, i, o, u}, then A⋃B = {a, b, c, d, e, i, o, u}
Negative Infinity - the -∞ symbol is used to denote negative infinity.
This is used when the set continues without end in the negative direction on a number line. Example: (-∞, a]
Positive Infinity - the ∞ symbol is used to denote positive infinity.
This is used when the set continues without end in the positive direction on a number line. Example: [ a, ∞)
Another way to write a set is using set-builder notation.
This is useful when the individual elements are not easily written.
Solving an equation
1. Remove any parentheses by using the Distributive Property. 2. If fractions or decimals remain, multiply each term by the least common denominator (LCD) of all the fractions. 3. Simplify each side, if possible. 4. Add or subtract terms on both sides of the equation to get all the variable terms on one side of the equation. 5. Add or subtract number terms on both sides of the equation to get all the number terms on the other side of the equation. 6. Multiply or divide both sides of the equation to get the variable alone on one side of the equation. 7. Simplify the solution. 8. Check your solution.
Solving and Equation Using the Addition Property of Equality
1.) Add or subtract the same number from both sides of the equation to get the variable on one side of the equation by itself. - if a number is being added to x, use subtraction - if a number is being subtracted from x, use addition 2.) Simplify, if needed, by combining like terms. 3.) Check your solution.
To solve equation of the form Ax + B = C, when a, b, and c are real numbers, do the following:
1.) Get the variable term alone on one side of the equation. Use the Addition Property of Equality to add or subtract the same number from both sides. 2.) Get the variable alone on one side of the equation. Use the Multiplication Property of Equality to multiply or divide both sides of the equation by the coefficient of the variable. - if the coefficient is a fraction, multiply both sides by its reciprocal. 3.) Simplify, if needed, by combining like terms. 4.) Check your solution.
Solving a Formula for a Specified Variable: To solve a formula or an equation for a specified variable, use the same steps that are used to solve an equation EXCEPT treat the specified variable as the only variable in the equation and treat the other variables as if they are numbers.
1.) Identify the variable you are solving for. 2.) Remove any parentheses by using the Distributive Property. 3.) If fractions or decimals remain, multiply each term by the least common denominator (LCD) of all the fractions. 4.) Simplify each side if possible. 5.) Add or subtract terms on both sides of the equation to get all terms containing the specified variable on one side of the equation. 6.) Add or subtract terms on both sides of the equation to get all the terms not containing the specified variable on the other side of the equation. 7.) Multiply or divide both sides of the equation to get the specified variable alone on one side of the equation. 8.) Simplify the solution.
Solving an Equation Using the Multiplication Property of Equality
1.) Multiply or divide both sides of the equation by the same number to get the variable x on a side of the equation by itself. - if x is being multiplied by a number, use division - if x is being divided by a number, use multiplication 2.) Simplify, if needed, by combining like terms 3.) Check your solution
To Graph a Linear Equality
1.) Plot the boundary point, which is the point that separates the solutions and the non-solutions. a.) If the boundary point is a solution (≥ or ≤), use a closed circle: <-----● ●-----> b.) If the boundary point is not a solution (> or <), use an open circle: <-----○ ○-----> 2.) Shade all numbers to the side of the boundary point that contains the solutions to the inequality.
Translating Words to Equations: When solving word problems, it is important to break down the problem to understand it.
1.) Read the word problem carefully to get an overview. 2.) Determine what information you will need to solve the problem. 3.) Draw a sketch or make a table. Label it with the known information.
To Determine if a Given Value is a Solution:
1.) Substitute the given value into the equation. 2.) Simplify each side of the equation according to the order of operations. 3.) If the result is a true statement, then that value is a solution.
To solve an equation, reverse operations are often needed.
1.) The reverse operation of addition is subtraction. 2.) The reverse operation of subtraction is addition.
The inequality x > 3 means that x could have the value of any number greater than 3.
5 > 3 = true statement, 5 is a solution to x > 3 0 > 3 = not a true statement, 0 is not a solution to x > 3
An inequality is a statement that shows the relationship between any two real numbers that are not equal. "NOTE: Inequalities can also be used to express the relationship between a variable and a number.
< "is less than" > "is greater than" ≤ "is less than or equal to" ≥ "is greater than or equal to"
"Set A is the set of all natural numbers greater than 4"
A = {x | x is a natural number greater than 4} A = {x | x ϵ N and x > 4}
A = {a, e, i, o, u}
A = {x | x is a vowel in the alphabet}
Set
A set is a collection of like objects called elements. The symbol ϵ means "is an element of the set." For example, Ringo ϵ Beatles.
*NOTE
Addition and subtraction "undo" each other, meaning that adding and subtracting the same number result in no change.
Solving Equations with Decimals Using the LCD
An equation containing decimals can be solved in a similar way. You can multiply both sides of the equation by an appropriate power of 10 to eliminate the decimal numbers and work only with integer coefficients.
*NOTE
An equation may have one solution, more than one solution, or no solution.
Infinity - an infinite set is a set whose elements cannot be counted.
An example of this is the set of real numbers. The set of real numbers increases without bound to the right and decreases without bound to the left on a number line.
Examples of Formulas
C = 2πr... the formula for finding the circumference of a circle P = 2l + 2w... the formula for finding the perimeter I = Prt... the formula for finding simple interest
There are several ways to write a set:
C = {1, 2, 3, 4, 5, 6} C = {x | x is a natural number less than 7} C = {x | x ϵ N and x < 7} C = {x | x ϵ N and x ≥ 6}
When solving an equation, simplify both sides of the equation whenever possible.
Combining like terms on both sides of the equation will make it easier to work with.
1.) Understand the problem. 2.) Choose a variable to represent the unknown quantity. 3.) Write an expression to represent each unknown quantity in terms of the variable. Look for key words to help you translate the words into algebraic symbols and expressions. 4.) Use a given relationship in the problem or an appropriate formula to write an equation. 5.) Write the equation.
EXAMPLE: One-third of a number is fourteen. 1/3 x n = 14 1/3n = 14 Five more than six times a number is three hundred five. 5 + 6 x n = 305 5 + 6n = 305 The larger of two numbers is three more than twice the smaller number. The sum of the numbers is thirty-nine. Larger number = 3 + 2s s + 3 + 2s = 39
If A = {multiples of 3} and B = {multiples of 4}, then A⊆B
False... A = {3, 6, 9, 12...} and B = {4, 8, 12, 16...}
Solving equations with parentheses
For all real numbers a, b, and c, a(b + c) = ab + ac
The Multiplication Property of Equality
If both sides of an equation are multiplied by the same non-zero number, the solution does not change. a, b, and c with c ≠ 0, if a = b, then ca = cb
*NOTE
If the decimals are tenths, multiply by 10; if the decimals are hundredths, multiply by 100, etc...
The Addition Property of Equality
If the same number is added to both sides of an equation, the results on both sides are equal in value. That is, adding the same number to both sides of an equation, does not change the solution. If a = b, then a + c = b + c
Remember, if the sign of the variable term is positive, us subtraction to reverse the operation.
If the sign of the variable term is negative, use addition to reverse the operation.
*NOTE
If you know the value of x, then the order of operation tells us to multiply before adding. When trying to solve for x, we must "undo" this. That is, we must add (or subtract) first, then multiply (or divide).
Solving Equations in the Form ax + b = cx + d
In some cases, a term with a variable may appear on both sides of the equation. In these cases, it is necessary first to rewrite the equation so that all the terms containing the variable appear on one side of the equation. We do this by adding or subtracting one of the variable terms from both sides.
To make the calculations a little easier, we can perform an extra step that will allow us to rewrite the given equation with fractions as an equivalent equation that does not contain fractions.
To make the process of solving equations with fractions easier, multiply both sides of the equation by the least common denominator (LCD) of all the fractions contained in the equation. Then use the Distributive Property to multiply each term in the equation by the LCD. If done correctly, all fractions will change into integers.
If A = {odd numbers} and B = {integers}, then A⊆B
True... A = {1, 3, 5, 7...} and B = {...-2, -1, 0, 1, 2, 3, ...}
Intersections and Unions
Two operations that are used with sets are the intersection and the union.
Solving an Inequality
Use the same procedure to solve an inequality that is used to solve an equation, EXCEPT the direction of an inequality must be reversed if you multiply or divide both sides of the inequality by a negative number.
Subsets
When all of the elements of one set are contained in another set, the smaller set is a subset of the larger set.
Solving an Inequality
When we solve an inequality, we are finding all the values that make the inequality true.
Set X is the set of all natural numbers between 4 and 10. Set Y is the set of all natural numbers between 2 and 7, inclusive.
X = {5, 6, 7, 8, 9} Y = {2, 3, 4, 5, 6, 7}
If the elements in a set can be counted, the set is called a finite set. Otherwise, the set is infinite.
Y = {2, 3, 4, 5, 6, 7} is a finite set N = {1, 2, 3, 4, 5...} is an infinite set
Solving equations in the form Ax + B = C
You must use both the Addition Property of Equality and the Multiplication Property of Equality together.
Solution
a solution of an equation is the number(s) that, when substituted for the variable(s), makes the equation true.
Equations with an Infinite Number of Solutions
an equation has an infinite number of solutions if the equation is always true, no matter the value of x. The solution of such equations is all real numbers.
Equations with No Solutions
an equation has no solution if there is no value of x that makes the equation true. The symbol used to show no solution is Ø.
Linear Inequality
contains a single variable on either side of the inequality symbol. x > 3 3x - 5 < 8 4x - 6 ≥ 8x + 12
Equivalent Equation
equations that have exactly the same solutions.
Variable
is a letter or symbol that represents an unknown quantity.
Equation
is a mathematical statement that two expressions are equal. All equations contain an equal sign ( = )
Graph of an Inequality
is a picture that represents all of the solutions of the inequality. We graph inequalities by shading in all possible solutions to the inequality on a number line.
Formula
is an equation in which variables are used to describe a relationship.
Linear Equation
is an equation that can be written in the form Ax + B = C, where A, B, and C are real numbers and A ≠ 0
Contradiction
is an equation that is false for all values. That is, when different values appear on both sides of the equal sign, we call the equation a contradiction.
Identity
is an equation that is true for all values. That is, when the same values appear on both sides of the equal sign, we call the equation an identity.
Solution of an Inequality
is any number that makes the inequality true.