Equation
An equation is a mathematical statement that two expressions are equal.We can solve equations.We can simplify expressions. Expressions do not contain an equal sign (=).
Solve an Equation
To solve an equation means to find the value or values of the variable that make the equation true.
Properties
Let a, b, and c be real numbers.If a=b, then a+c=b+c and a-c=b-c.If a=b and c is not zero, then ac=bc and a/c=b/c.
Linear Equation
A linear equation in one variable is an equation that can be written in the form ax+b=0 where a and b are real number and a is not zero.
Process
If there are terms in parentheses, begin by distributing.Combine like terms on each side of the equal sign.Get the terms with the variables on one side of the equal sign and the constants on the other side using the addition and subtraction properties of equality.Solve for the variable using the multiplication or division property of equality.Check the solution in the original equation.
No Solution
An equation has no solution if a contradiction results from the solution process.We say the solution set is the empty set.
Infinite Number of Solutions
An equation has an infinite number of solutions if an identity results from the solution process.We say the solution set is the set all real numbers.
Process
Read the problem carefully. Then read it again. Draw a picture, if applicable.Identify what you are being asked to find. Define the variable; that is, assign a variable to the unknown quantity. Also,If there are other unknown quantities, define them in terms of the variable.Label the picture with the variable and other unknowns as well as with any given information in the problem.Translate from English to math. Some suggestions for doing so areRestate the problem in your own words.Read and think of the problem in "small parts."Make a chart to separate these "small parts" of the problem to help you translate to mathematical terms.Write an equation in English, then translate it to an algebraic expression.Solve the equation.Interpret the meaning of the solution as it relates to the problem. If there are other unknowns, find them. State the answer in a complete sentence. Be sure your answer makes sense in the context of the problem.
Simple Interest
I=PRTI=interest earned (simple)P=principal (initial amount invested)R=annual interest rate (expressed as a decimal)T=amount of time the money is invested (in years)
And / Intersection
Take the overlap
Or / Union
Take everything (but only take things that overlap once)
Linear Inequality in One Variable
A linear inequality in one variable can be written in the formax+b<c,ax+b<=c,ax+b>c, orax+b>=c,where a, b, and c are real numbers and a is not zero.
Process
The process is mostly the same as for solving linear equations in one variable. However,when we multiply or divide both sides of the inequality by a negative number we must reverse the direction of the inequality symbol.
Solutions
Write in inequality notationGraph on a number lineWrite in set notationWrite in interval notation
Compound Inequality
Three parts: all operations must be performed on each of the three parts
Treat all variables you are not solving for as constants.