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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
x = -1
8 - 2 (x + 1) = 9 + x
x = 5
6 (x - 5) + 4 = 4 (x - 4)
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.
-4 = x
1/4x - 2/3 = 5/12x
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
Identity
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.
0 = 22
There is no solution to this equation
7x + 13 = 13 + 7x
There is an infinite number of solutions to this equation
-21x - 21x
There is an infinite number of solutions to this equation
0 = 68
There is no solution to this equation
1/8x + 1 = 1/8x + 1
There is an infinite number of solutions to this equation
20x - 100 = 20x - 100
There is an infinite number of solutions to this equation
9 = 40
There is no solution to this equation
Formula
An equation that relates two or more quantities
Examples of Formulas
C = 2πr... the formula for finding the circumference of a circle
x = 3
ax + b = c
ta = v
Solve for x.
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.
EXAMPLE: 4 < 6
First, you must examine what occurs when you add, subtract, multiply, or divide both sides of the inequality by a positive number.