Algebra 1

To graph a linear inequality, first, plot coordinates as if the equation was not an inequality. Next, decide the type of line to connect the points with. Is the line solid or dashed? Lastly, shade in the appropriate region using a test point to help if necessary.

Solving inequalities with addition and subtraction is very similar to solving regular equations. The goal is to figure out what the variable represents. As far as inequalities go, the variable now represents a section of possible numbers. In the case of addition and subtraction the inequality will remain the same throughout the entire problem. Remember that what we do to one side of the inequality, we must also do to the other.

Solving inequalities with multiplication and division is also similar to solving regular equation. However, we must be careful to and remember to flip the inequality whenever we multiply or divide by a negative number. Remember that what we do to one side of the inequality, we must also do to the other.

To solve multi-step inequalities, it is always appropriate to start with any simple addition or subtraction and then move on to multiplication and division, but keep in mind that parentheses can affect your approach. Our goal is to solve for the given variable so we must isolate that to one side of the inequality. The hardest part is keeping track of the orientation of the inequality. Remember to flip it any time a negative number is multiplies or divided to both sides.

A set is a collection or a group. Think of grouping together items such as ingredients or colors. The notation for a set is a list separated by commas and surrounded by brackets. For example, here is a set of cardinal directions, {North, South, East, West}. Ellipses are use to show that a set continues on.

Compound inequalities are multiple inequalities combined together with the words “and” or “or”. When combined with or, the answer is everything that both inequalities represent. When combine with and, the solution is only the overlap of the inequalities.

We know that an absolute value represents how far a number is from zero and is always positive. Therefore, when an absolute value is present in an inequality, we must remember that negative numbers must also be taken into account.

Union represents all of the members of the different sets and intersection represents the common numbers between each set. For example, if I had two different sets, one representing colors of my house floor, {tan, brown, white}, and another representing he colors of my house walls, {blue, green, white}. The union would be all of the colors, {tan, brown, white, blue, green}, and the intersection would only be one color, {white}.

A relation is a set of ordered pairs. This relation can be displayed as an ordered pair, a table, graphing, and mapping. The inverse relation is when the coordinates of the original relation are flipped or switch so the x-coordinate becomes the y-coordinate and vice versa. When graphed together, you can see that the inverse relation is a reflection of the original relation over the line y=x.

The domain of a relation is the x values and the range of a relation is the y values. With that being said, if you have an inverse relation, then the domain and range are also switched so that the domain represents the y values and the range represents the x values.

Equation relations are about utilizing our skills to understand a given equation based on the many tools used to represent a relation. Sometimes you will be given a domain or range and other times you will be asked to solve for those attributes.

A function is a relation where each input has a single output. In other words, each element in the domain has exactly one element that it is paired with in the range. You can also think of it as each x-coordinate having only one y-coordinate that it goes with. If a relation has an x value that can be matched with two different y values, then the relation is not a function.

An arithmetic sequence is a sequence in which there is a common difference between one term and the next. That difference can be either positive or negative.

Direct variation represents an equation in the form of y=kx where the number k, which is also known as the constant of variation, does not equal zero. Think of the equation in terms of y=mx+b. Direct variation is an equation that has no y-intercept, so there is no b. Therefore; the slope is the constant of variation.

In order to write a linear equation in slope-intercept form we must first find the slope and the y-intercept of the equation. Based on the information given and the context of the problem, the slope may be given and then we would need to figure out the y-intercept or the y-intercept may be given and then we would need to find the slope. At times, the given information could consist of only points and both slope and y-intercept would need to be calculated. A good rule of thumb is to always start by writing out y=mx+b and then filling in from there.

Point slope form requires knowing the slope and one ordered pair. The for point slope form is y-y*sub 1=m(x-x*sub1), where *sub 1 stands for subscript 1. Slope is again characterized by the variable m and the known point is (x*sub 1, y*sub 1). The y intercept is not necessary in point slope form.

Parallel lines are lines with the same slope and different y-intercepts. In point slope form, the slopes are still the same, but the points must belong to separate lines since parallel lines never intersect. Perpendicular lines are lines where the slopes are opposite reciprocals of one another, and that is key in point slope form as well.

A scatter plot is a graph of ordered pairs or points that show a relationship between different data sets. The correlation or relationship between two points can be positive meaning that as the overall x value increases, the y values increase. Correlation can be negative meaning as the x values increase, the y values tend to decrease. Data sets can also show no correlation meaning that the relationship between them appears random.

A system of equations is two or more equations together. The solution to a system would be an ordered pair that is holds true for each equation in the system. To graph a system of equations, graph each individual equation in the same coordinate system so that they all appear on the same graph. If the lines intersect, then there is exactly one solution and it is both independent and consistent. If they do not intersect, then they are parallel lines, have no solution, and it’s inconsistent. If the equations represent the same line the there are infinitely many solutions, and is dependent and consistent.

Substitution is simply plugging in the variables value. After substituting, the problem can be simplified. A lot of times that simplified problem will lead to a solution.

Equations can be added and subtracted to other equations. This is a useful tool when solving a system. By adding or subtracting, entire variables can total up to equal zero, which will remove that variable from the problem, allowing you to solve for any remaining variables. Once you solve for the remaining variables, you will be able to return to the original equation and solve for the variable that was canceled out, and therefore, solving the entire system.

Many times adding or subtracting equations will still not allow you to solve a system because the variable you wish to cancel out cannot equal zero. However, by multiplying one or both equations, you may manipulate the coefficients of the variables you wish to cancel and then add or subtract. Just remember when multiplying an equation, you must multiply by the same number on both sides of the equals sign.

A system of inequalities is very similar to a system of equations. One of the easiest ways to understand how a system of inequalities works is to graph each inequality individually being sure to shade in the appropriate region for each. The regions that intersect represent the solution to the system.

A monomial is one term: such as a variable, number, or product of numbers and variables. There are properties such as Product of Powers, Power of a Product, and Power of a Power that help in simplifying and solving a monomials that are multiplied together.

In order to divide monomials there are other properties that we can utilize to simplify and solve. There is the Quotient of Powers, Power of a Quotient, Negative Exponent definition, and the Zero Exponent definition that will be used when dividing monomials.

A polynomial is both a single monomial and a combination of monomials that are added together. In other words, a monomial is always a polynomial, but a polynomial does not have to be a monomial. A polynomial could be a binomial, which is two monomials added together. A polynomial could also be a trinomial, which is three monomials added together, and so on. A degree of a monomial is the sum of all the exponents that are with variables. A degree of a polynomial is the greatest degree out of all the monomials that combine to make the polynomial itself.

Adding and subtracting polynomials might seem a little intimidating. The first step is to combine like terms. Combining like terms is adding together the same types of terms with the same degrees. For example, 4x can me combined with -9x, but not with 3y since the variables are different. Also, 4x cannot be combined with 6x^2 even though the variables are the same and the reasoning is because the degrees are different.

Multiplying a monomial with a polynomial is a process completed by using distribution. The monomial must be multiplied to each individual monomial within the polynomial and then simplified.

Distribution is used to also find the products of two polynomials. For this case, each monomial in the first polynomial must be multiplied to each monomial in the second polynomial. A trick to multiplying two binomials is using the term FOIL. Start by multiplying the First monomials, then the Outside monomials, next the Inside, and then followed by the Last.

A prime number is an integer that is greater than 1 where the only factor are 1 and itself. For example, 13 is a prime number because 1 and 13 are the only factors of the product 13. A composite number is an integer that is not prime, but also greater than 1. So a composite has to have more than 2 factors. Prime factorization is a whole number written as a product of factors that are only prime. For example, 21 would be written as 3 times 7 because both 3 and 7 are prime and they multiply to become 21.

The greatest common factor, or GCF for short, is the greatest integer that divides both of the given numbers. To find the GCF of those given numbers, start by writing out all of the factors of each number. The given numbers will have at least one factor in common and whichever is the greatest is the GCF.

Think of doing the reverse of distribution. To start factoring polynomials, we first must find the prime factorization of each monomial within the polynomial that we are factoring. Next, we discover the GCF of all the monomials and remove it from each of them. Finally, the polynomial will be written as a product of the GCF and whatever was left of the monomials.

Factoring basic polynomials, we simply did the reverse of distribution. So for factoring x^2+bx+c, we are going to reverse FOIL.

Factoring ax^2+bx+c is a little more complex than factoring ax^2+bx+c or the basic polynomials. However, by grouping and simplifying and following a few tips, factoring will be easier than ever!

If you are given a^2 –b^2 we know that the expression is equal to (a+b)(a-b). If you are given (a+b)^2, then that expression equals (a+b)(a+b). And if you are given (a-b)^2, we have (a-b)(a-b).

A quadratic function is in the form of y=ax^2+bx+c. Quadratic functions will form a parabola which will look like either a U or an upside down U, in other words a smile and a frown. The vertex is known as the maximum or minimum and is either the highest or the lowest point on the graph depending on whether the parabola opens up or down. Each parabola will have an axis of symmetry and will look like a reflection over that line.

A solution to a quadratic equation is the roots of the equation which occurs when y=0. We also know this to be the x-intercept. A quadratic equation can have 0 solutions, 1 solution, or 2 solutions.

The quadratic formula is a formula used to find the exact roots of a quadratic equation. The formula, in words, is the opposite of b plus or minus the square root of b squared minus 4ac all over 2a. [–b +or- sqrt(b^2-4ac)]/2a = x. Solving is a matter of plugging in values for a, b and c, and then calculating for x.

The discriminant is b^2-4ac. You may recognize this as the expression within the square root of the quadratic equation. The value of the discriminant helps determine how many solutions there are to the given quadratic equation. If the discriminant is positive, then there are 2 solutions. If it is equal to 0, then there is 1 solution. And if the discriminant is negative, then there are no solutions.

An exponential function is a function in the form of y=a^x where a is grater than 0 and not equal to one. This is different from all of the problems we have done so far because we have a variable as an exponent. Growth occurs when a is greater than 1 and decay occurs when a is less than 1.

Geometric sequences are similar to arithmetic sequences that we learned about in a previous chapter. However, instead of there being a common difference between the terms, there is a common ratio that is not equal to 1 or 0.