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College Prep Math Review Prepared by Professor L. Blanchette Real Number System and the Real Number Line: All real numbers correspond to points on the real number line. The Real Number System is ordered. If a real number lies to the right of another real number on the number line, it is larger. Likewise, if it lies to the left of a number on the number line, it is smaller. EXAMPLE: -3 > -7 because -3 lies to the right of -7 on the number line. Absolute value gives the distance a number lies from zero on the number line. The absolute value of a number is always non-negative. EXAMPLE: The absolute value of 11 is the same as the absolute value of -11; both numbers are 11 units from zero on the number line. All Real Numbers can be subdivided into two categories: Rational and Irrational Numbers Rational Numbers: In decimal form, rational numbers are decimal numbers that either repeat or terminate. In fractional form, rational numbers are numbers that can be written in the form of integer over integer. Irrational Numbers: In decimal form, irrational numbers are decimal numbers that do NOT repeat and do NOT terminate. In fractional form, irrational numbers are numbers that cannot be written in the form of integer over integer. Natural Numbers: Natural numbers are also called counting numbers. The smallest natural number is 1. Natural Numbers are a subset of the Real Numbers. The set of Natural Numbers = {1, 2, 3, 4, 5 . . .} Whole Numbers: Whole numbers include all the natural numbers plus zero. Integers: Integers include the natural numbers, their negatives, and zero. The set of Integers = {. . . , -3, -2, -1, 0, 1, 2, 3 . . .} If a radical number can be simplified so as to rewrite without the radical symbol, that radical number is considered a rational number. Otherwise, it will be either irrational or not real. Signed Numbers and Mathematical Operations: To add signed numbers: If the numbers have the same sign, add the digits and keep the sign. If the numbers have different signs, subtract the digits and take the sign of the number with the larger absolute value. In algebra, subtracting is referred to as simply adding a negative. EXAMPLE: 13 - 27 = 13 + (-27) = -14 The middle step is a mental step and should not be written. To multiply signed numbers: If you multiply two numbers with the same sign, the result will be a positive number. If you multiply one positive and one negative number, the result will be a negative number. Translating an English phrase or statement into a Mathematical expression or equation: Look for key words. Addition words: sum, plus, more than, greater than, added to, increased, raised, total Subtraction words: difference, minus, less than, subtracted from, decreased, dropped, fell Multiplication words: product, times, multiplied by, twice (means multiplied by two) Division words: quotient, divided by, split evenly, shared equally Equality: is, is equal to, the result is Inequality: is greater than, is greater than or equal to, is less than, is less than or equal to, is not equal to NOTE: In algebra, division is reflected as a fraction bar line rather than the divide symbol. An unknown number can be represented by a letter. This letter is called a variable. If the problem has more than one unknown, clearly assign the variable to one unknown and then express the other unknown in terms of that variable if at all possible. Always state clearly what the variable represents. Expressions vs. Equations: Algebraic expressions do NOT contain an equal sign (or an inequality symbol.) Algebraic expressions can be simplified. Perform the necessary mathematical operations, working inside the grouping symbols first, distributing as necessary, and combining like terms. Algebraic expressions can also be evaluated. If the value for the variable is provided, replace the variable with that value (using parenthesis) and simplify. The answer will be a number value. Equations, on the other hand, can be solved. This means we can find the value for the variable that makes the original statement true. This number value is called the solution. Solving Linear Equations: Think of an equation as a statement of balance. When you solve an equation, remember to keep the balance. That is, whatever you do to one side of the equation must be also done to the other side. You can multiply both sides of an equation by any non-zero number, you can divide both sides by any non-zero number, and you can add any number to both sides, or subtract any number from both sides. The key is to do the SAME thing to both sides. Your goal is to isolate the variable. You want to have the letter on one side of the equation by itself. The solution will be on the other side. Solving Inequalities: Inequalities: Inequalities can be solved in the same way that equations are solved. However, there is one important new rule. If you multiply or divide both sides of an inequality by a NEGATIVE number, you must change the direction of the inequality symbol! Also, keep in mind that the answer to an inequality is an interval, not a point. You can express the solution graphically, as an inequality statement, with set builder notation, or with interval notation. Exponents and Roots: Exponents: Exponents indicate the number of factors of that base. Remember, the exponent is on the number, or grouping symbol, immediately to its left. The sign of the number or expression is NOT affected by the exponent unless that sign is inside a grouping symbol and the exponent is outside that same grouping symbol. Review the exponent rules and be able to simplify algebraic fractions involving exponents. Roots: Radical symbols are used to indicate roots. The index indicates which root must be found. If a radical symbol does not show an index, that index is the invisible 2. So the 2nd root is indicated by using the square root symbol. The square root symbol is used to find the principal (positive) root only. Simplify all radical expressions using radical rules. Factoring: Factoring an expression means to rewrite it as a product. In factoring, remember to factor completely: final factors should be prime factors. Look for a GCF first! The GCF is the greatest common factor. If there is a number that divides evenly into all terms, use the biggest such number. If every term has the same variable, that variable is part of the GCF as well. Use the smallest exponent on that variable. 4 Terms: If the polynomial has 4 terms try Factor-by-Grouping. 2 Terms: If the polynomial has only 2 terms, check for special patterns: Difference of Two Squares, Sum or Difference of Two Cubes. 3 Terms: If the polynomial has 3 terms, check for the special pattern: Perfect Square Trinomial. Else, consider the two cases: (1) The leading coefficient is 1, or (2) the leading coefficient is NOT 1. (1) When the leading coefficient is 1, if the polynomial is factorable, it will be relatively "easy" to rewrite using( )( ) form and finding the numbers that multiply to the product number while also adding to the sum number. (2) When the leading coefficient is NOT 1, you can either Guess and Check, or Split and Group. Review these methods and be able to use either method. NOTE: Not all polynomials are factorable. If the polynomial cannot be factored, it is Prime. Solving Quadratic Equations: Quadratic equations can be solved by factoring, by the square root property (if it has the form x2 =a), by completing-the-square, or by using the quadratic formula. Not all quadratic equations are factorable. If they factor nicely, this is generally the preferred method. Both completing-the-square and the quadratic formula are methods that work for solving all quadratic equations. Word Problems: Be sure to read the entire problem through carefully. Determine what question needs to be answered. Make a sketch or use a box (table) set-up to organize the information and visualize the situation. List all the pertinent known facts. List the unknown facts. Assign a variable to the unknown. Use only one variable. Express other unknown facts in terms of the one variable. Be specific and label your sketch or box. Is there a formula that applies? If so, write this down and use it. You may need to make your own formula that applies to this particular problem. Write a verbal statement that ties the key facts together. Use mathematical terms. Now translate this statement into an algebraic equation. It is important to solve word problems algebraically. Solve the equation. Check for errors or unreasonable results. Be sure to answer the original question! Use units in your answer. Do not use any answers that are unreasonable. Geometric Problems: Know the basic geometry facts. These will be assumed knowledge. You should know how to find the perimeter and area of a rectangle, the perimeter and area of a triangle, the circumference and area of a circle, and the volume of rectangular solids. In addition, you need to know basic angle facts, and facts about parallel lines. Review these. Be comfortable with them. NOTE: To maximize the area of a rectangular enclosed region, use a square. Graphing: The rectangular coordinate system is also called the Cartesian coordinate system. Solutions to linear equations in two variables can be represented by points in the coordinate plane. The line connecting these points represents all such possible solutions.