Accuplacer Elementary Algebra Review Hennepin Technical College Placement Testing for Success Page 1 Overview The Elementary Algebra section of ACCUPLACER contains 12 multiple choice Algebra questions that are similar to material seen in a Pre-Algebra or Algebra I pre-college course. A calculator is provided by the computer on questions where its use would be beneficial. On other questions, solving the problem using scratch paper may be necessary. Expect to see the following concepts covered on this portion of the test: Operations with integers and rational numbers, computation with integers and negative rational numbers, absolute values, and ordering. Operations with algebraic expressions that can be simplified using formulas and expressions, adding and subtracting monomials and polynomials, multiplying and dividing monomials and polynomials, simplifying with positive rational roots and exponents, simplifying algebraic fractions, and factoring algebraic expressions. Operations that require solving equations, inequalities, and word problems, solving linear equations and inequalities, using factoring to solve quadratic equations, solving word problems and written phrases using algebraic concepts, and geometric reasoning and graphing. Testing Tips Use resources provided such as scratch paper or the calculator to solve the problem. DO NOT attempt to solve problems only in your head. Start the solving process by writing down the formula or mathematic rule associated with solving the particular problem. For equations, check your answer by substituting the answer back into the original problem. Make an educated guess if you are unsure of the answer. Hennepin Technical College Placement Testing for Success Page 2 Algebra Tips Test takers should be familiar with the following concepts. For specific practice exercises using these concepts, please utilize the resources listed at the end of this guide. Understanding a number line Multiplying two binomials Adding and subtracting negative numbers Using proportions to solve problems Using exponents Combining like terms Finding a square root Evaluating expressions Writing algebraic expressions Solving linear equations Using parentheses in algebraic expressions Solving systems of equations Evaluating formulas Practice Questions Order of Operations Grouping Symbols Order of Operations is GEMDAS Exponents Multiply & Divide (Left Right) Add & Subtract (Left Right) 42 52 1. Simplify: 372 2. Simplify: 3 + 2(5) – | –7 | 3. Simplify: (4 5)2 Scientific Notation When writing a large number in When writing a very small number in scientific notation, the exponent will scientific notation, the exponent will be be positive. negative. 1. Write 0.00000000000523 in scientific notation. 2. Write 6.021011 in expanded form. 3. Multiply 3103 5106 . Write final answer in scientific notation. 6109 4. Divide . Write final answer in scientific notation. 3104 Hennepin Technical College Placement Testing for Success Page 3 Evaluate Expressions Find the value of each expression when x = 3, y = – 4, and z = 2. 5x z 1. Evaluate: xyz – 4z 2. Evaluate: xy Linear Equations in One Variable 1. Solve for x : 6x – 48 = 6 2. Solve for x : 50 – x – (3x + 2) = 0 Formulas Use the order of operations in reverse order to isolate the designated variable. 1. Solve for T : PV = nRT 2. Solve for x : y = hx + 4x Word Problems 1. One number is 5 more than twice another number. The sum of the numbers is 35. Find the numbers. 2. Sheila bought burgers and fries for her children and some friends. The burgers cost $2.05 each and the fries are $.85 each. She bought a total of 14 items, for a total cost of $19.10. How many of each did she buy? Inequalities 1. Solve and graph: 2x – 7 > 3 2. Solve and graph: 3(x – 4) – (x + 1) < –12 Hennepin Technical College Placement Testing for Success Page 4 Exponents & Polynomials Simplify and write answers with positive exponents. 24x4 32x3 16x2 1. 3x2 5x 6 5x2 4x 4 2. 3. (5a 6)2 8x2 Factoring 1. Factor: x2 + 5x – 6 2. Factor: 2x2 + 4x – 16 3. Factor: 4x2 – 36 4. Factor: 49y2 + 84y + 36 Quadratic Equations 1. Solve: 4a2 + 9a + 2 = 0 2. Solve: (3x + 2)2 = 16 Rational Expressions To add or subtract: Find a common denominator (factor the denominators only), then add or subtract numerators, keep the common denominator, and then reduce if possible. To multiply or divide: Factor all expressions in numerators and denominators, and then divide out common factors. When dividing rational expressions, remember to multiply by the reciprocal. 4 3a 16 x2 x2 2x 8 1. Add: 2. Divide: 2a 2 a 2 a x2 2x 8 4 x2 Hennepin Technical College Placement Testing for Success Page 5 Graphing Graph each equation on the coordinate axis. 1. 3x – 2y = 6 2. x = –3 3. y = 2 2 4. y x 5 3 Systems of Equations Systems of two linear equations are two lines graphed on the coordinate plane that meet at a single point. This point is the solution to the system and satisfies both equations. Solve the following systems of equations. 2x 3y 12 2x 3y 4 1. 2. x 2y 9 y 2x 4 Radicals Perform the indicated operations. Rationalize any denominators. All variables represent positive numbers. 1. 8 10 2. 2 18 5 32 7 162 12 15 3. 4. 2 3 5 23 3 4 2 18 40 Hennepin Technical College Placement Testing for Success Page 6 Answers Order of Operations 1. = 147 2. 3 + 2(5) – | –7 | = 6 3. = –9 Scientific Notation 1. 0.00000000000523 = 5.23 10–12 2. = 602,000,000,000 3. = 15 109 = 1.5 1010 4. = 2 105 42 52 372 Evaluate Expressions (4 5)2 5(3) 2 13 1. xyz – 4z = (3)(–4)(2) – (4)(2) = –24 – 8 = –32 2. (3)(4) 12 Linear Equations in One Variable 1. 6x – 48 = 6 x = 9 2. 50 – x – (3x + 2) = 0 x = 12 Formulas6.02 1011 3 PV 6 y 1. PV = nRT 3 10 510 T 2. y = hx + 4x x nR h 4 6109 4 310 Hennepin Technical College Placement Testing for Success Page 7 Answers (continued) Word Problems 24x4 32x3 16x2 1. x3 x=2 “another 5x 6 number”5x2 4x and 4 2x + 5 = “one number.” Remember, sum means to add.(5a 6Since)2 x + 2x + 5 = 35, then x = 10 which is “another number”8x2 and 2x + 5 = 25 which is “one number.” 2. Let x = the number of burgers and 14 – x = the number of fries. To get the total amount of money spent, multiply the number of items by the cost of the item. 2.05x = the total dollars spent on burgers and 0.85 (14 – x) = the total dollars spent on fries. The equation is: 2.05x + 0.85 (14 – x) = 19.10. Solving the equation, x = 6. Hence, she bought 6 burgers and 8 fries. Inequalities Solve inequalities the same as equations with one exception. When both sides are multiplied or divided by a negative number, remember to switch the inequality symbol. 1. 2x – 7 > 3 x > 5 1 2. 3(x – 4) – (x + 1) < –12 x < 2 Exponents & Polynomials 1. = 8x2 x 2 2. = 3x2 4x 2 3. = 5a 65a 6 = 25a2 60a 36 Factoring 1. x2 + 5x – 6 = (x + 6)(x – 1) 2. 2x2 + 4x – 16 = 2(x – 2)(x + 4) 3. 4x2 – 36 = 4(x + 3)(x – 3) 4. 49y2 + 84y + 36 = (7y + 6)(7y + 6) = (7y + 6)2 Hennepin Technical College Placement Testing for Success Page 8 Answers (continued) Quadratic Equations 1 1. 4a2 + 9a + 2 = 0 (4a + 1)(a + 2) = 0 a = and a = –2 4 2 2. (3x + 2)2 = 16 Take square root of each side: 3x + 2 = 4 3x = –2 4 x = , –2 3 Rational Expressions 2 y x 5 4 3a 4(a) 3a(2) 4a 6a 10a 5 1. 3 = = = = = 2(a 1) a(a 1) 2a(a 1) 2a(a 1) 2a(a 1) 2a(a 1) (a 1) (4 x)(4 x) (2 x)(2 x) (4 x) (2 x) 2. = = = (1)(1) = 1 (x 4)(x 2) (x 4)(x 2) (x 2) (x 4) (4 x) (2 x) Recall that 1 and 1 (x 4) (x 2) Graphing 3 3 1. 3x – 2y = 6 y x 3 (slope is and y-intercept is –3) 2 2 2. x = –3 (vertical line through –3) 3. y = 2 (horizontal line through 2) 2 4. (slope is and y-intercept is 5) 3 4 3a 16 x2 x2 2x 8 2a 2 a 2 a x2 2x 8 4 x2 Hennepin Technical College Placement Testing for Success Page 9 Systems of Equations Solve the following systems of equations. 1. Solution: (3, 6) 2. Solution: (1, 2) Radicals 1. = 80 = 16 5 = 4 5 2. = 2 9 2 5 16 2 7 81 2 = 6 2 20 2 63 2 = 49 2 180 1 1 3. = = = 720 4 2 4. = 6 9 8 6 15 6 20 4 = 6(3) 7 6 20(2) = 22 7 6 2x 3y 12 2x 3y 4 x 2y 9 y 2x 4 8 10 2 18 5 32 7 162 12 15 2 3 5 23 3 4 2 18 40 Hennepin Technical College Placement Testing for Success Page 10 .