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Basic Algebra Practice Test 1. Exponents and integers: Problem type 2 Evaluate. This is (9)( 9) = 81 This is (4)(4) = 16 2. Exponents and signed fractions Evaluate. Write your answers as fractions. 3. Exponents and order of operations Evaluate. 74 4. Evaluation of a linear expression in two variables Evaluate the expression when = 3 and = 6. +2 (3) + 2(6) (3) 12 3 12 15 5. Distributive property: Advanced Use the distributive property to remove the parentheses. 6. Combining like terms in a quadratic expression Simplify the following expression. and 7. are the only like terms Writing a mathematical expression Write an algebraic expression to answer the question below. If one notebook costs dollars, what is the cost, in dollars, of 8 notebooks? The cost of 8 notebooks will be 8 times the cost of one notebook. Since one notebook costs dollars, this will be 8 . 8. Solving a two-step equation with integers Solve for . 7 + 4 = 23 Simplify your answer as much as possible. 7 + 4 = 23 +7 +7 4 = 16 9. Solving a two-step equation with signed fractions Solve for . Simplify your answer as much as possible. Multiply both sides by the common denominator 12. 10. Solving a linear equation with several occurrences of the variable: Problem type 2 Solve for . Simplify your answer as much as possible. Multiply both sides by the common denominator 10. 11. Solving a linear equation with several occurrences of the variable: Problem type 3 Solve for . Simplify your answer as much as possible. Distribute 12. Solving a linear inequality: Problem type 3 Solve the inequality for . Simplify your answer as much as possible. Reverse the inequality sign when dividing by a negative number. 13. Graphing a linear inequality on the number line Graph the inequality below on the number line. The inequality sign is a , so use a filled in circle at . shade to the left of 3. 5 4 3 2 1 0 1 2 3 4 is any number less than 3, so 5 14. Algebraic symbol manipulation: Problem type 2 Solve the following equation for . Whenever an equation has a fraction, multiply both sides by the denominator. 15. Solving a fraction word problem using a simple linear equation The yearbook club had a meeting. The meeting had 24 people, which is two-thirds of the club. How many people are in the club? This is the same as 24 people is two-thirds of the club The word is will translate as an equal sign and of will translate as multiplication. You can use for people. Multiply both sides by the denominator There are 36 people in the club. 16. Solving a word problem using a linear equation: Problem type 2 Customers of a phone company can choose between two service plans for long distance calls. The first plan has no monthly fee but charges $0.14 for each minute of calls. The second plan has a $15 monthly fee and charges an additional $0.09 for each minute of calls. For how many minutes of calls will the costs of the two plans be equal? We want: Cost of first plan = Cost of second plan Cost of first plan = $0.14 for each minute times the number of minutes Cost of second plan = $15 monthly fee + $0.09 for each minute times the number of minutes You can use for minutes. . . . . . . . . . . The costs of the two plans are equal for 300 minutes. 17. Set builder and interval notation Graph the set on the number line. Then, write the set using interval notation. is between the endpoints 3 and 2. The endpoint has a sign, the inequality sign there has an equal, so use a filled in circle at this endpoint. The endpoint 2 has a < sign, so use an open circle. Shade between these. • 5 4 3 2 1 0 1 2 3 4 5 The left endpoint of the interval is 3 and the inequality sign there has an equal, so use a bracket for the left of the interval notation. The right endpoint of the interval is 2 and the inequality sign there does not have an equal, so use a parenthesis there. [3, 2) 18. Function tables The function is defined by the following rule. Complete the function table. ( ) 5 14 1 2 0 1 2 7 3 10 19. Domain and range from ordered pairs Suppose that the relation is defined as follows. , , , , , Give the domain and range of . Write your answers using set notation. The domain is the set of first values. Domain: {3, 1, 6} The range is the set of second values listed only once. Range: { , 1} 20. Graphing a line given the x- and y-intercepts Graph the line whose -intercept is 3 and whose -intercept is 2. y -intercept x -intercept 21. Graphing a line given its equation in slope-intercept form Graph the line. y You can make a table of values for points to plot. 0 1 2 x 3 1 1 22. Graphing a line through a given point with a given slope Graph the line with slope 2 passing through the point (1, 3). y The slope is 2 = x 1 2 . To plot another point go either 2 in the -direction and 1 in the -direction or go 2 in the -direction and 1 in the direction from (1, 3). 23. Graphing a vertical or horizontal line Graph the line = 1. y x The graph of = a number will always be a vertical line that crosses the -axis at the number. 24. Finding x- and y-intercepts of a line given the equation in standard form Find the -intercept and -intercept of the line given by the equation. 4 +9 =8 For -intercept, let = 0 and solve for . 4 + 9(0) = 8 4 =8 =2 -intercept = 2 For -intercept, let = 0 and solve for . 4(0) + 9 = 8 9 =8 -intercept = 25. Finding slope given two points on the line Find the slope of the line passing through the points (3, 4) and (8, 3). slo e slo e 26. Writing an equation of a line given the y-intercept and a point Write an equation of the line below. y slo e -intercept = = 2 1 2 x The equation of a line is = + 27. Writing the equations of vertical and horizontal lines through a given point Write equations for the vertical and horizontal lines passing through the point (9, 3). Vertical line: = 9 Horizontal line: = 3 28. Evaluating expressions with exponents of zero Evaluate the expressions. Anything to the zero power is 1 1 29. Writing a negative number without a negative exponent Rewrite the following without an exponent. A negative exponent means reciprocal. This is the same as moving between the numerator and denominator and changing the sign of the exponent. 30. Multiplying monomials Multiply. Simplify your answer as much as possible. This can be rewritten as When multiplying the same base, add the exponents. 31. Product rule of exponents in a multivariate monomial Simplify. Use only positive exponents in your answer. This can be rewritten as When multiplying the same base, add the exponents. Move the variables that have a negative exponent to the denominator. 32. Quotients of expressions involving exponents Simplify. There is one factor in the numerator and 6 factors in the denominator. The one in the numerator cancels with one in the denominator leaving 5 in the denominator. There are 5 factors in the numerator and 4 factors in the denominator. The 4 in the denominator cancel with 4 in the numerator leaving one in the numerator. 33. Power rule with negative exponents: Problem type 2 Simplify. Write your answer using only positive exponents. When a product is raised to a power, raise each factor to the power. When a power is raised to a power, multiply the exponents. Move the variables that have a negative exponent to the denominator. 34. Using the power, product, and quotient rules to simplify expressions with negative exponents Simplify. Write your answer using only positive exponents. This problem may be easier if you get rid of the negative exponents, first. The 3 exponent means reciprocal. Move the rest of the variables with negative exponents between the numerator and denominator. Raise everything in the first parentheses to the 3 power. There are 3 factors in the denominator and 4 factors in the numerator. The 3 in the denominator cancel with 3 in the numerator leaving one in the numerator. Multiply the s by adding the exponents. 35. Ordering numbers with negative exponents Order the expressions by choosing <, >, or =. The negative exponent means reciprocal. These can be rewritten and then the order can be determined. < > < 36. Simplifying a polynomial expression Simplify. Change the subtraction to add the opposite. To find the opposite of a polynomial, change the sign of each term in the polynomial. The parentheses can be taken out and like terms combined. 37. Multiplying a monomial and a polynomial: Problem type 2 Multiply. Simplify your answer as much as possible. Distribute the multiply by to everything in the parentheses. 38. Multiplying binomials: Problem type 3 Multiply. Simplify your answer. Use FOIL to multiply. Multiply the firsts, multiply the outsides, multiply the insides, and multiply the lasts. 39. Degree and leading coefficient of a polynomial in one variable What are the leading coefficient and degree of the polynomial? The degree is the largest exponent and the leading coefficient is the coefficient of the term with the largest degree. Leading coefficient = 7 Degree = 5 40. Factoring a quadratic with leading coefficient 1 Factor. To factor a quadratic with leading coefficient 1, find two numbers that multiply to give the last term that also add to give the coefficient of the middle term. Find two numbers that multiply to give 36 and that add to give 9. These numbers are 12 and 3. These are the two numbers in the factors. The quadratic will factor as ( + 12)( 3) 41. Factoring a quadratic with leading coefficient greater than 1 Factor. To factor a quadratic with leading coefficient greater than 1, find two factors that multiply to give the first term and two numbers that multiply to give the last term. This may not always give the correct answer, so you must check the answer. Two factors that multiply to give are 5 and . Two numbers that multiply to give 4 are 2 and 2. This MAY factor as (5 + 2)( + 2) Check this by multiplying. This checks. The factors are (5 + 2)( + 2) 42. Greatest common factor of two monomials Find the greatest common factor of the two expressions and Simplify your answer as much as possible. Find the greatest common factor of the numbers then find the greatest common factor of the variables. The greatest common factor of 10 and 15 is 5. The greatest common factor of the variables will be the smallest number of times it is used in both. 43. Factoring out a monomial from a polynomial: Problem type 2 Factor the following expression. The common monomial factor will be the greatest common factor of the two terms. The greatest common factor is . Each term can be written as a multiple of the greatest common factor. Use the distributive property to rewrite this. 44. Factoring a product of a quadratic trinomial and a monomial Factor completely. First, factor out a common monomial from the polynomial. Factor the remaining polynomial. This is a quadratic with leading coefficient 1. Find two numbers whose product is 25 and whose sum is 10. The two numbers are 5 and 5. Use these to write the factors. Remember the 3 factor. 45. Multiplying rational expressions: Problem type 1 Multiply. Simplify your answer as much as possible. Cancel common factors before multiplying. 2 goes into both 4 and 2. 5 goes into both 5 and 25. There is an in both the numerator and denominator, these will cancel. There is one in the numerator and 3 in the denominator; these will cancel leaving 2 in the denominator. There is another in the numerator and 2 in the denominator; these will cancel leaving 1 in the denominator. Now multiply. 46. Adding rational expressions with common denominators Subtract. Simplify your answer as much as possible. Change the subtraction to add the opposite. To find the opposite of a fraction, find the opposite of the numerator. To do this, change the sign of each term in the numerator. Now combine like terms in the numerator and keep the common denominator. 47. Dividing rational expressions: Problem type 2 Divide. Simplify your answer as much as possible. To divide, change to multiply by the reciprocal. Now factor everything. There is a common factor of 1 in the numerator and denominator that will cancel. There is also a common factor of + 3 that will cancel. Multiply these for the final answer. 48. Simplifying a ratio of polynomials: Problem type 1 Simplify. Factor everything. Now cancel common factors 49. Solving a rational equation that simplifies to a linear equation: Problem type 1 Solve for . Simplify your answer as much as possible. Whenever an equation has a fraction, multiply both sides by the denominator. The s on the left will cancel. Divide both sides by 9. This will simplify. 50. Finding the roots of a quadratic equation with leading coefficient 1 Solve for . First, factor the quadratic. Find two numbers whose product is 12 and whose sum is 4. These numbers are 6 and 4. Set each factor equal to zero. 6 = 0 and + 2 = 0 Solve these. = 6 and = 2