Algebra Basics

What is a double inequality?

An inequality where the value of the expression is indicated between two quantities.

The distributive property of multiplication also applies when the expression inside grouping symbols is preceded by what? What effect does this have on the signs of each term inside the group?

An opposite or minus sign/ the effect of changing the signs of each term inside the group.

What is a set?

An organized collection of objects, such as numbers.

What is set builder notation?

Another method of writing a set of elements where the vertical line is read as "such that".

What is a polynomial?

Any algebraic expression with more than one term(monomial).

What is a numerical expression?

Any combination of numbers and operations.

How do you simplify an expression?

By combining like terms: 1. Identify the terms that are like terms (the ones with the same variable factors raised to the same exponents or with the same constant terms) 2. Add or subtract the coefficients as indicated. 3. Retain the common variables.

When an expression must be evaluated repeatedly on a graphing calculator, what is a more efficient way of doing so?

By entering the expression on the Y screen rather than on the home screen. Then, each time we store a value for the value for the variable, we can obtain the corresponding value of the expression.

How can we relate the value of an expression for a given value of the variable?

By writing the relationship as an ordered pair in the form of (value of the variable, value of the expression).

Describe how a graph can indicate special cases of inequalities.

Special cases arise when the graphs of the left and right sides of an inequality are parallel, and the inequality symbol determines if a solution set is empty or R in these cases. When the graph of an inequality shows parallel lines, the graph of the side of the inequality that is indicated by the inequality symbol to be the lesser value is always below the graph of the side of the inequality having the higher value. In this case the solution set is R. When the graph of inequality shows parallel lines, but the inequality is the opposite of what it should be from above, the inequality is false. The solution set is therefore empty set.

What are natural numbers?

1 through 9. Excluding 0.

Describe two properties of equations that are essential in solving equations algebraically.

1. Addition Property of Equations: For real numbers a,b, and c, the equations a = b and a + c = b + c are equivalent equations. Adding the same number to both sides of the equation results in an equivalent equation. Because subtraction is defined in terms of addition, we can also subtract the same number from both sides of an equation. 2. Multiplication Property of Equations: For real numbers a, b, and c, c not being equal to 0, the equations a = b and ac = bc are equivalent equations. Allows us to multiply or divide (because multiplication can be defined in terms of division) by the same nonzero number and obtain an equivalent equation.

What are 2 ways in which equivalent equations can be produced?

1. By exchanging the two sides of the equation. 2. By replacing any expression with an equivalent expression.

What is the solving routine for Linear equations in one variable?

1. Clear fractions by multiplying both sides by the LCD of all the fractions. 2. Use the Distributive Property to simplify both sides. a. Remove grouping symbols. b. Combine like terms. 3. If a variable term appears on both sides, use the Addition Property of Equations to eliminate the variable term from one side. 4. Use the Addition Property of Equations to isolate the variable term. 5. Use the Multiplication Property of Equations to isolate the variable itself. 6. Solve the resulting equation by inspection and verify the solution.

What is the Greek letter epsilon used for in algebra?

Stands for 'is an element of' and is used to describe a quantity as being a part of a set of elements. The symbol also represents the lower case Greek letter Epsilon.

What are the primary uses for mathematical properties?

To describe the structure of the set of real numbers and for deriving new rules. Basic properties play an important role in all future study of algebra for they are the foundation of more advanced algebra.

If there is no solution to an equation, its solution set is called an _______ and is indicated how? If every real number is a solution to an equation, its solution set is ___, the set of all real numbers.

empty set/0 with a slash through it: ∅/R

When adding algebraic expressions, only ____ terms can be combined.

like or similar.

Highlighting the point corresponding to an ordered pair is called _______ the point.

plotting

After you enter an expression on the Y screen, you have to keep in mind that as you trace the coordinates of the expressions graph, the first coordinate is the value of the _____ and the second coordinate is the value of the _____.

variable/expression

If a coefficient is not specifically indicated, it is understood to be ____________.

1 or -1

How can using the graphing method to estimate the solution to an equation lead us to suspect that an equation is either a identity or contradiction?

1. If the graphs of each expression don't seem to intersect (look parallel), then the equation is probably a contradiction and has no solution. 2. If the graphs of each expression seem to coincide (there appears to only be one line on the graph) then the equation is probably an identity.

What are the rules of multiplying and dividing signed numbers?

1. If the two numbers have like signs, their product or quotient is always positive. 2. If the two numbers have unlike signs, their product or quotient is always negative.

Describe the order of operations

1. Perform operations within grouping symbols performing the ones within the inner most grouping symbols first. 2. Perform operations involving exponents and square roots. 3. Perform operations of multiplication and division from left to right. 4. Perform operations of addition and subtraction from left to right.

What are the two steps of simplifying an algebraic expression?

1. Remove grouping symbols. 2. Combine like terms.

What is an equation?

A mathematical statement that two quantities are equal.

What is an algebraic expression?

A collection of numbers and variables separated by normal operators; addition, subtraction, multiplication, and division. This collection of numbers and variables represents the value of something.

What is a directed or signed number?

A number that has negative or positive sign.

What is an irrational number?

A number that when written in decimal form, does not have a terminating decimal or block of repeating digits. There are endless non-repeating digits. Irrational numbers can written as decimals such as these but not as fractions. They are also often the result of taking the square root of a number.

What is a linear inequality in one variable?

An inequality that can be written as Ax + B > 0, where A is not equal to 0. A similar definition can be stated for <, greater than or equal to, and less than or equal to.

The term algebra is derived from the Persian ______________.

Aljabr

What are real numbers?

All the natural numbers, whole numbers, integers, and rational, and irrational numbers combined.

What is a term?

An algebraic expression connected only by the operations of multiplication and division.

What is a monomial?

An algebraic expression with only one term

What is a trinomial?

An algebraic expression with only three terms separated by plus or minus signs.

What is a binomial?

An algebraic expression with only two terms separated by plus or minus signs.

What is a coordinate Plane?

An arrangement where a horizontal number line and vertical number intersect. It is used to visualize the relationship between sets of numbers. Conventionally, we represent the x-values along a horizontal number line called the x-axis and the y-values along a vertical number line called the y-axis.

What is a contradiction?

An equation that is false for all replacements of the variable.

What is an identity? Give a few examples.

An equation that is true for any permissible replacement of the variable. Examples would an expression in the form of the associative property of multiplication or the commutative property of addition: 6(3x)=18x and y+3=3+y.

What is a conditional equation?

An equation that may be true or false depending on the replacement of the variable(s).

How is a variable isolated in an equation?

Apply any inverse operations to both sides of the equation until the variable is isolated. Make sure to use the same operation on each side of the equation or the balance of the equation will be destroyed and the solution of the equation will be incorrect.

Why is it particularly important to read the symbol -x as "the opposite of x," and not "negative x"?

Because -x is not always necessarily a negative number.

Why would you want to graph an inequality on a number line?

Because it can help to visualize the solutions which are quite often very many or even infinite in number.

Pairs of numbers are called ordered pairs because why?

Because the order in which the coordinates are written is significant.

If an equation involves fractions, we can usually obtain a simpler equation how?

By eliminating the fractions. The Multiplication Property of Equations allows us to multiply both sides of the equation by any nonzero number. If we multiply both sides by the LCD of all the fractions and then simplify, the resulting equation will have no fractions. We call this process clearing fractions. Just make sure you multiply every term on both sides of the equation by the LCD.

What is an equivalent equation?

Equations that have the same solutions.

What are equivalent fractions?

Fractions that have the same decimal value.

What is the Multiplication property of -1

For any real number a, a(-1) = -1a = -a. Entitles us to write the symbol -a as -1a if we want to. This property is useful in simplifying expressions.

What are nested grouping symbols?

Grouping symbols inside of grouping symbols are to be worked with first when using the order of operations.

What is Algebra?

It is essentially a way to reduce a math problem down to a small set of symbols.

What is the solution?

It is the number or numbers which, when used as the value of a variable or variables, makes the statement true.

What is the origin of a coordinate plane?

It is the point at which the x-axis and y-axis intersect.

What is the absolute value of a number?

It is the unsigned or undirected value of a number.

What is a reciprocal?

It is what is obtained when you interchange a fractions numerator with its denominator.

What are other names for a coordinate plane?

Rectangular coordinate system or the Cartesian coordinate system.

What are similar or like terms?

Terms that are similar or like because they are both constant terms or they have the same literal factors raised to the same powers.

What properties are commonly used to combine like terms?

The Commutative and Associative properties of addition.

What property is commonly used to simplify expressions by removing parentheses?

The Distributive Property of multiplication.

What do algebraic expressions allow us to do?

They allow us to communicate ideas and generalize (simplify) big ideas.

Describe the quadrants of a coordinate plane.

They are numbered 1 through 4 starting from the upper right hand gong counterclockwise.

What is a coefficient?

The numerical factor in a term.

What is the Least Common Denominator?

The smallest number that can be divided evenly by each denominator in a fraction problem.

An algebraic expression does not have a specific value until what?

Until the variables in the expression have been replaced with numbers.

When estimating the solution of an equation, the x value can be found from a table in a graphing calculator how?

When each Y value is equal, the corresponding x value is the solution to the equation.

When is a number prime factored?

When the is written as a product whose factors are all prime numbers.

How is raising a negative base that IS NOT enclosed in parentheses to a power different than raising a negative base that IS in parentheses to a power?

When you raise the base that IS NOT in parentheses to a power, only one factor of the base will have a negative value while the rest will be positive. When you raise a negative base that IS in parentheses, each factor of the base is negative.

Plotting the points corresponding to the ordered pairs of a set are called ________the set.

graphing

The more formal name for opposite is _________?

additive inverse

Real numbers are a part of a larger system of numbers called what?

complex numbers

The two numbers in an ordered pair that describe a point are called ______

coordinates

As with equations, we can exchange the left and right sides of inequalities. However, we must take care to change the _____ of the inequality symbol along with the flipped expressions.

direction

When the coefficient of a variable term is an integer, it is usually easiest to ____ both sides of the equation by the coefficient. When the coefficient is a fraction, _____both sides by the reciprocal of the fraction is an efficient way to isolate the variable.

divide/multiplying

The procedure for solving a linear inequality is nearly ______ to the procedure for solving a linear equation.

identical

Using parentheses to denote multiplication is also called what?

implied multiplication

The variable must be ________________on one side of the equation.

isolated.

Although algebraic methods lead to exact solution, the solution should be what?

verified as a check against errors

We usually do not write an improper fractions as a ________ in algebra.

mixed number

A more formal name for reciprocal is ____________?

multiplicative inverse

In the first forms of the Distributive Property, we ______ to do what? In the second forms, we ______ to do what?

multiply/to convert an indicated product a(b+c) into an indicated sum ab + ac./factor/to convert an indicated sum ab + ac into an indicated product a(b+c).

One raised to any number equals _______

one

The number lines in a coordinate plane divide the plane they are on into ______

quadrants.

Since terms are formed using only multiplication and division, plus and minus signs _____________ terms and make expressions with more than one term.

separate

It is often convenient to describe the solution(s) of an equation with a set called the __________.

solution set

Use the Addition Property of Equations to eliminate a ____; use the Multiplication Property of Equations to eliminate a ______.

term/coefficient

What does the graph of an expression represent?

the infinitely many values of the variable and the corresponding values of the expression.

When we solve double inequalities algebraically, we use the Addition and Multiplication Properties of Inequalities on all _______ of the inequality to isolate the variable.

three parts

The opposite of zero is___________?

zero

Zero raised to any number equals _______

zero

In addition the additive identity is ___ and in multiplication the multiplicative identity is ____.

0/1

What is the solving routine for linear inequalities in one variable?

1. Clear fractions by multiplying both sides by the LCD of all the fractions. 2. Use the Distributive Property to simplify both sides. a. Remove grouping symbols. b. Combine like terms. 3. If a variable term appears on both sides, use the Addition Property of Inequalities to eliminate the variable term from one side. 4. Use the Addition Property of Inequalities to isolate the variable term. 5. Use the Multiplication Property of Inequalities to isolate the variable itself. (If you multiply or divide each side of the inequality by a negative number, then you have to flip the inequality symbol. 6. The resulting inequality describes the solutions of the original inequality. The solutions can also be given by a solution set or described by a number line graph.

Describe how you estimate solutions of inequalities graphically with a calculator.

1. Enter the left-side expression y1 as Y1 and the right -side expression y2 as Y2. 2. Produce the graph of Y1 and Y2. 3. Trace to the point of intersection and estimate the x-coordinate. For less than or equal to and greater than or equal to inequalities, this x-coordinate is included as a solution. 4. Trace the graph of y1 to determine the following: a. If the inequality has the form y1 < y2 or y1 is less than or equal to y2, find where the graph of y1 is below the graph of y2. b. If the inequality has the form y1 > y2 or y1 is greater than or equal to y2, find where the graph of y1 is above the graph of y2. 5. Write an inequality that describes the solutions. (The solutions can also be given as a solution set, or they can be described with a number line graph.)

How do you estimate the solutions of a double inequality by graphing?

1. Graph all three expressions and trace to estimate points of intersection. 2. Then observe where the graph of y2 is between the graphs of y1 and y3. 3. The solutions are the x-coordinates of the points in that interval.

How do you test the truth of an equation on a graphing calculator?

1. Store the value for x. 2. Enter the entire equation in and hit enter. 3. 1 is true and 0 is false.

Describe the four methods used for evaluating an algebraic expression with a graphing calculator.

1. Store the value of the variable and enter the expression on the home screen. 2. Enter the expression on the Y screen, store the value of the variable on the home screen, and retrieve the y-value. 3. Enter the expression on the Y screen, and produce the graph. Trace to the point whose x-coordinate is given and read the y-coordinate which is the value of the expression. 4. Enter the expression on the Y screen and produce a table. Scroll to the given x-value and read the value of the expression in the Y column.

Describe the accuracy of the graphing method of estimating solutions to equations.

A graph only allows you to see as close as you can with your eyes the points that are plotted on a graph. Since it's impossible to see with just your eyes the exact point at which two expressions intersect or the exact point of any ordered pair for that matter, this can only be considered an estimation of the solution. Therefore, graphs are merely suggestive until a proposed solution is verified by other means.

What is an ordered pair?

A pair of numbers that describe a point on a coordinate plane. The first number indicates horizontal direction on the x-axis and the second number indicates vertical direction on the y-axis.

What is a multiple?

A product of any quantity and an integer. Also can be defined as a number that can be divided by another number without a remainder.

What is a constant term?

A term with no variable.

What is a Linear Equation in one variable?

A type of an equation that can be written in the form Ax + B = 0, where A and B are real numbers and A is not equal to 0. It is "linear"because it creates a straight line when plotted on a graph.

When dealing with negative numbers, what is an easy way to determine the sign of the product? This follows from what logic?

An even number of minus signs multiplied will result in a positive product while an odd number of minus signs will result in a negative product. This follows from the logic that negative numbers can be paired up to give positive results and a negative number times a positive number results in a negative product.

Why is there no end to the number of possible pairs and points of a graphed expression?

Because the expression can be evaluated for any infinite number of real number values of its variable.

What is a literal number and a variable?

Both are a letter that stands for a number.

What are rational numbers?

Fractions where the numerator and denominator are integers as long as the bottom integer (divisor) is not 0.

What is the definition of subtraction?

For any real numbers a and b, a - b = a + (-b)

What is the Addition Property of Inequalities?

For real numbers a, b, and c, the inequalities a < b and a + c < b + c are equivalent. A similar result holds for >, less than or equal to, and greater than or equal to. Allows us to add the same number to both sides of an inequality as well as subtract the same number from both sides.

What is the Multiplication Property of Inequalities?

For real numbers a, b, and c: 1. If c > 0, then the inequalities a < b and ac < bc are equivalent. 2. If c < 0, then the inequalities a < b and ac > bc are equivalent. A similar result holds for >, less than or equal to, and greater than or equal to. We can multiply (or divide) both sides of an equation by a negative number, but we must reverse the direction of the inequality symbol.

What are the rules of adding signed numbers and subtracting signed numbers?

If you are adding: If they both have like signs, then add their absolute values and use the sign common to both. If they both have opposite signs, subtract their absolute values and keep the sign of the largest numerical value. If you are subtracting: Change the sign of the subtrahend (the number being subtracted) and then use the rules for adding.

If solving a linear equation leads to an equivalent equation with no variable, then what does this mean?

It means that the solution to the equation is a special case where: 1. If the resulting equation is false, then the original equation is a contradiction, and the solution set is empty (there are no solutions that satisfy the original equation) 2. If the resulting equation is true, the original equation is an identity, and the solution set is R. Every real number will work as the solution for the original equation.

What does it mean when you obtain an equivalent inequality with no variable?

It means that this answer is a special case: 1. If the inequality is true, then the original inequality is true for all real numbers. The solution set is R. 2. If the inequality is false, there is no number for which the original inequality is true. The solution set is empty.

What does solving an equation algebraically mean?

It means using algebraic operations to produce equivalent equations until we obtain one that can be solved by inspection. If an equation can be written in the form x = number, then the number is the solution. Thus our strategy is to keep reducing the equation to simpler and simpler equivalent equations until the variable is isolated to one side of the equation.

What does simplifying an algebraic expression mean?

It means writing an equivalent expression that contains no grouping symbols and that has no terms that can be combined.

What is the roster method?

The method of writing a set by listing its elements. It uses brackets to group the elements:

What is the Least Common Multiple?

The smallest number that is a multiple of two or more given numbers.

How can you estimate the solution of an equation?

You estimate by graphing. 1. Graph each side of the equation. 2. Trace to the point of intersection of the graphs. 3. The x-coordinate of the point of intersection is the estimated solution of the equation. (The y-coordinate is the corresponding value of each side of the equation.)

What is a plane?

You can think of a plane as a flat, unbounded surface.

When graphing an inequality on a number line, we use the symbols ___at an interval to indicate that the number at that particular interval is included and the symbols ____at an interval to indicate that the number at that particular interval is not included.

[ and ]/( and )

Properties can be used to rewrite numerical expressions so they are easier to perform__________?

mentally

When we use the Addition Property of Equations, the sign of the number we add to both sides is the ______ of the sign of the ____ we are trying to eliminate. The term is thereby eliminated because the sum of the two opposites is ____. When we use the Multiplication Property of Equations, the sign of the number by which we multiply or divide both sides is the ____ as that of the ______ we are trying to eliminate. The coefficient is thereby eliminated because we are dividing out the common factor.

opposite/term/0/same/coefficient