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Level 198

Linear Equations & Inequalities in One Variable

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A mathematical sentence that contains an equals sign
1.) Read the word problem carefully to get an overview.
When solving word problems, it is important to break down the problem to understand it.
5.) Write the equation.
an alphabetic character representing a number, called the value, which is either arbitrary or not fully specified or unknown. It is usually a letter like x or y.
The value or values that make an equation or inequality true.
An equation may have one solution, more than one solution, or no solution.
1.) Substitute the given value into the equation.
To Determine if a Given Value is a Solution:
Linear Equation
A polynomial equation of the first degree whose graph is a line.
Equivalent Equation
equations that have exactly the same solutions.
In this form, the number is a solution of the equation.
By following certain procedures, we can often transform an equation into a simpler equivalent equation that has the form of x = some number.
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.
1.) The reverse operation of addition is subtraction.
To solve an equation, reverse operations are often needed.
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.
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
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.
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.
Solving equations in the form Ax + B = C
You must use both the Addition Property of Equality and the Multiplication Property of Equality together.
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.
If the sign of the variable term is negative, use addition to reverse the operation.
Remember, if the sign of the variable term is positive, us subtraction to reverse the operation.
Solving equations with parentheses
For all real numbers a, b, and c, a(b + c) = ab + ac
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.
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.
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.
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.
Two statements are in blank if and only if they always have opposite truth values
The product of any number and one is that number.
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.
An equation that relates two or more quantities
Examples of Formulas
C = 2πr... the formula for finding the circumference of a circle
linear inequality
contains a single variable on either side of the inequality symbol.
solution of an inequality
Any number that makes the inequality true
5 > 3 = true statement, 5 is a solution to x > 3
The inequality x > 3 means that x could have the value of any number greater than 3.
Graph of an Inequality
Graph that shows all the solutions of the inequality
To Graph a Linear Equality
1.) Plot the boundary point, which is the point that separates the solutions and the non-solutions.
Solving an Inequality
When we solve an inequality, we are finding all the values that make the inequality true.
A _ is a well-defined collection of objects.
Natural numbers
A group of numbers that include 1,2, 3,.....
X = {5, 6, 7, 8, 9}
Set Y is the set of all natural numbers between 2 and 7, inclusive.
Y = {2, 3, 4, 5, 6, 7} is a finite set
If the elements in a set can be counted, the set is called a finite set. Otherwise, the set is infinite.
Another way to write a set is using set-builder notation.
This is useful when the individual elements are not easily written.
Set A is the set of all natural numbers greater than 4
A = {x | x is a natural number greater than 4}
A = {a, e, i, o, u}
A = {x | x is a vowel in the alphabet}
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.
The notation { Ø } means "a set whose only element is the empty set."
When writing the notation for an empty set, only use one of the symbols -- either { } or Ø, but not both.
There are several ways to write a set:
C = {1, 2, 3, 4, 5, 6}
Intersections and Unions
Two operations that are used with sets are the intersection and the union.
Set Intersection
The intersection of two sets is the set of all the elements that are common to both sets.
Set Union
The union of two sets is the set of every element that is in either or both sets.
When all of the elements of one set are contained in another set, the smaller set is a subset of the larger set.
B = {letters of the alphabet}
Set A is a subset of B. this can be written as A⊆B.
If A = {odd numbers} and B = {integers}, then A⊆B
True... A = {1, 3, 5, 7...} and B = {...-2, -1, 0, 1, 2, 3, ...}
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...}
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].
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.
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, ∞)
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]