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Elementary Algebra Review

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Elementary Algebra Review Placement Test Review Materials for 1 To The Student This workbook will provide a review of some of the skills tested on the COMPASS placement test. Skills covered in this workbook will be used on the Algebra section of the placement test. This workbook is a review and is not intended to provide detailed instruction. There may be skills reviewed that you will not completely understand. Just like there will be math problems on the placement tests you may not know how to do. Remember, the purpose of this review is to allow you to “brush-up” on some of the math skills you have learned previously. The purpose is not to teach new skills, to teach the COMPASS placement test or to allow you to skip basic classes that you really need. The Algebra test will assess you r mastery of operations using signed numbers and evaluating numerical and algebraic expressions. This test will also assess your knowledge of equations solving skills used to solve 1-variable equations and inequalities, solve verbal problems, apply formulas, solve a formula fro a particular variable, and fined “k” before you find the answer. You will also be tested on your knowledge of exponents, scientific notation, and radical expressions, as well as operations on polynomials. Furthermore, other elementary algebra skills that will be on the test include factoring polynomials, and using factoring to solve quadratic equations and simplify, multiply, and divide rational expressions. 2 Table of Contents Evaluating Expressions Using Order of Operations 4 Simplifying Complex Fractions 5 Evaluating Algebraic Expressions 6 Using Formulas 8 Solving Equations Solving 1-Variable Equations 9 Solving 1-Variable Inequalities 12 Writing Verbal Expressions as Equations and Algebraic Expressions 13 Solving an Equation for a Particular Variable 15 Solving Problems that Involve a Second Variable 16 Using Exponents Using the Rules for Exponents to Simplify 17 Using Scientific Notation 19 Simplifying Polynomials Adding and Subtracting Polynomials 20 Multiplying Polynomials 21 Using Factoring Factoring Polynomials 22 Solving Quadratic Equations Using Factoring 26 Applying Quadratic Equations 28 Simplifying Rational Expressions 29 Multiplying and Dividing Rational Expressions 30 Using Radicals Simplifying Radical Expressions 31 Adding, Subtracting, Multiplying and Dividing Radical Expressions 33 Answer Key 34 3 4 Using Order of Operations Rules: 1. Do the operation inside the parentheses, braces, brackets, or absolute value first. If more than one, do the innermost parentheses first. 2. Do the exponents next. 3. Multiply and Divide from left to right. 4. Add and Subtract from left to right. Examples: 5 + 3(8 – 9) – 7 = 52 – 3(7 + 2) ÷ 27 = 5 + 3(8 + -9) + -7 = 52 + -3 (9) ÷ 27 = 5 + 3(-1) + -7 = 25 + -3(9) ÷ 27 = 2 + -7 = -5 25 + -27 ÷ 27 = 25 + -1 = 24 WATCH OUT! 3( 2) 5 In an expression like , the fraction bar acts like a grouping symbol. Simplify 7 2( 2) the numerator and denominator separately, and then divide. Practice: 1. Simplify: 8 ÷ 23 – 12 ÷ 22 2. Simplify: 2 · 42 – 6 ÷ 2 – 9 · 4 36( 6) 3. Evaluate: 14 – 27 ÷ (3 – 6)2 – 3 4. Evaluate: 4(2 5)2 5 Simplifying Complex Fractions Simplify the numerator and the denominator first, and then divide the numerator by the denominator. 3 9 15 18 33 33 2 33 Example: 4 10 20 20 20 1 5 5 2 20 5 50 2 2 2 Practice: 3 1 4 1. Simplify: 2. Simplify: 5 13 1 1 38 10 21 23 34 34 3. Evaluate: 4. Evaluate: 2 5 1 7 2 6 Evaluating Algebraic Expressions Substitute the value(s) for the variable(s). If the value is a negative number, use a parentheses around the number. This will help prevent mistakes with negative signs. Use order of operations to simplify. Examples: Evaluate: -4(a – 2b) – bc for a = 3, b = -2 and c = 4. -4(a – 2b) – bc = -4(3 – 2(-2)) – (-2)4 = -4(3 – (-4)) – (-8) = -4(3 + 4) + 8 = -4(7) + 8 = -28 + 8 = -20 Find the value of 3x2 – x + 2 when x = -2. 3x2 – x + 2 = 3(-2)2 –(-2) + 2 = 3(4) + 2 + 2 = 12 + 2 + 2 = 16 Practice: 21 1. Evaluate a2 – b2 when a = 4 and b = -3. 2. Evaluate: x( xy 4 x ) when x = -6 32 and y = 3. 11 3. Evaluate -2a2 + 3b – c when a = -3, 4. Evaluate: mn22 when m = 3 34 b = 2, and c = 4. and n = 8. 7 5. Find the value of –x5 + 4x2 -9x -7 6. Evaluate the following polynomial for for x = -1. y = -1. 3y2 – y + 5 8 Using Formulas Read the problem carefully. Identify the known and unknown variables. Write a formula that relates the variable in the problem Substitute the known values for the variables. Solve the equation and answer the question. Example: Find the length of weather stripping needed to put around a rectangular window that is 9 ft. wide and 22 ft. long. The perimeter of a rectangle is found by using the formula: P = 2L + 2W, L = length and W = width. Known variables: W = 9, L = 22 Formula: P = 2L + 2W Unknown variable: P P = 2(22) + 2(9) P = 44 + 18 P = 62 ft. Practice: 1. Find the length of aluminum framing needed to frame a picture that is 6 ft. by 4 ft. The formula for the perimeter of a rectangle is P = 2L + 2W, where L = length and W = width. 2. Find the approximate length of a rubber gasket needed to fit around a circular porthole that has a 20-inch diameter. The formula for the circumference of a circle is C = π · d, where d = diameter. Use π = 3.14. 9 3. A room 11 ft. by 15 ft. is to be carpeted. Find the number of square yards of carpet needed (9ft2 = 1 yd2). Round to the nearest tenth. 4. The surface area of a right circular cylinder can be found by using the formula: SA = 2πrh + 2πr2. Use the formula to find the surface area of an oil drum whose radius is 4 ft. and whose height is 5 ft. Use π = 3.14. 10 Solving 1-Variable Equations An equation is a mathematical sentence that involves an equal sign. Examples of equations in one variable are: 3x – 6 = 9 2x + 5x – 8 = 5x – 3 3(x – 2) = 5x + 6 The following steps are used to solve an equation: 1. Remove parentheses and combine like terms on each side of the “=”. 2. Use the addition property to get all terms with variables on one side of the equation and all other terms on the other side. (Remember, if you add a term to one side of the equation, then you must add the same term to the other side to keep the equation balanced or true. 3. Combine like terms on each side of the equation. 4. Use the multiplication or division property to solve for the variable. Remember to multiply or divide both sides in order to keep the equation balanced or true. 5. Check your solution for the variable by substituting into the original equation and simplifying. The result should be a true statement. Examples: Solve: 3x – 8 = 4 Solve: 3(x + 5) + 2x = 6x + 10 3x + -8 = 4 3x +15 +2x = 6x + 10 3x + -8 + 8 = 4 + 8 5x + 15 = 6x + 10 3x = 12 5x + -6x + 15 = 6x + -6x + 10 3x = 12 -1x + 15 = 10 3 3 -1x + 15 + -15 = 10 + -15 x = 4 -1x = -5 -1 -1 x = 5 Practice: 1. Solve: 5x – 5 = 3x + 7 2. Solve: 2x – 11 = -3x + 9 11 3. Solve: 5x + 7 + 3x = 47 4. Solve: 4y + 6 + y = 11 5 1 5. Solve: x 35 6. Solve: x 52 8 3 7. Solve: 2x – 3(2x – 5) = 7 – 2x 8. Solve: 3x + 4(2x – 7) = 9x – 22 12 Solving 1-Variable Inequalities To solve an inequality, you follow the same rules that you use for solving an equation except: When you multiply or divide both sides by a negative number, you must change the inequality sign. Examples: Solve: 2x + 6 > 4 Solve: -3x + 8 > -4 2x + 6 + (-6) > 4 + (-6) -3x + 8 + (-8) > -4 + (-8) 2x > -2 -3x > -12 2x > -2 -3x > -12 2 2 -3 -3 x > -1 x < 4 (did not change > sign) (did change > to < because I divided by a negative number) Practice: 1. -5x + 2 < -18 2. 3x – 2 > 13 3. Solve: 5x – 1 < -9 4. Solve: 2 – 7x < 16 13 Writing Verbal Expressions as Equations and Algebraic Expressions Words that mean add: Words that mean subtract: plus total minus reduced by sum greater than fewer difference between increased by more than decreased by less than Words that mean multiply: Words that mean divide: times product of quotient of divisor of at (sometimes) dividend ratio total(sometimes) parts Examples: Translate into an algebraic expression or equation: three more than twice a number 2n + 3 eight less than 1/3 of a number 1/3n – 8 four times a number increased by 6 4n + 6 the product of a number and 5 5n Translate into an equation or inequality. Then solve: Four more than three times a number equals 8. 3n + 4 = 8 3n + 4 + (-4) = 8 + (-4) 3n = 4 3n = 4 3 3 n = 4/3 The difference between twice a number and 3 is greater than 15. 2x – 3 > 15 2x + (-3) + 3 > 15 + 3 2x > 18 2x > 18 2 2 x > 9 14 Practice: 1. Translate “six less than the product of n and negative nine” into an algebraic expression: 2. Write “the quotient of ten and the sum of a number and 7 as an algebraic expression 3.
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