MATH PLACEMENT TEST STUDY GUIDE

Transcription

1 MATH PLACEMENT TEST STUDY GUIDE The study guide is a review of the topics covered by the Columbia College Math Placement Test The guide includes a sample test question for each topic The answers are given at the end Arithmetic Calculators may not be used for this test because part of the test involves knowledge of basic arithmetic a) Adding whole numbers You must know the sum of any two one-digit numbers: You must also know how to add larger whole numbers For example, to find the sum: : First the digits in the units column are added: = The is carried to the tens column, The digits in the tens column, including the carried are then added: = the is carried to the hundreds column, Finally the digits in the hundreds column, including the carried are added: ,to finish the sum:

2 Sample Question a) Find the sum: b) Multiplying whole numbers You must know the product of any two one-digit numbers: X You must also know how to do long multiplication For example to find the product: 4 5 : Sample Question b) Find the product: 57 6 a c) Lowest terms A fraction is an expression of the form b where a b are whole numbers The top number, a, is called the numerator the bottom number, b, is called a the denominator A fraction b is said to be in lowest terms if a b have no common 8 factor For example, the fraction is not in lowest terms because 4 is a factor (or divisor) of the numerator 8 is also a factor of the denominator : if we divide the 8 numerator the denominator by the common factor 4, the fraction will be expressed in lowest terms as Sample Question c)

3 04 Express 0 in lowest terms d) Multiplying fractions When a number x is multiplied by a number y, the result xy is called the product of x y Fractions are multiplied according to the rule: a c ac = b d bd 5 8 For example, the product of 8 5 expressed in lowest terms is: = = Note that in practice we may use cancellation The cancellation consists of dividing the top bottom by 5 (ie replacing 5 5 with 7 respectively) also dividing the top bottom by 4 (ie replacing 8 8 with 7 respectively) This is justified by the rules of arithmetic: = = = = = Sample Question d) 6 5 Express 5 7 in lowest terms a Dividing fractions If the numerator, a, the denominator, b, of a fraction b are both ac a multiplied by the same non-zero number c, the result bc equals b because ac a c a a a = = = bc b c b b Furthermore, if a fraction b is multiplied by its reciprocal, b a b ab a = = a, the product equals : b a ab It follows that dividing a fraction b by c a c another fraction d is equivalent to multiplying b by the reciprocal of d : a c a d ad = = b d b c bc e)

4 because: a a d a c = b = b c = b d c c d d d c = = = For example, Note that in practice we use cancellation: a b d c = a b d c Sample Question e) 45 Express 6 in lowest terms f) Adding subtracting fractions Fractions that have the same denominator are added or subtracted according to the rules: a c a + c a c = 5 + = b b b + = For example, a c a c + = a + c = (a + c) = + These rules make sense because: b b b b b b a c a c = a c = (a c) = b b b b b b When adding or subtracting fractions that have different denominators, the fractions must first be expressed as equivalent fractions with a common denominator, preferably the lowest common denominator,(the smallest number that both denominators divide into), so that the previous rules may be applied = + = + = For example: a b c b = b Sample Question f)

5 4 8 + Express 5 70 in lowest terms g) Adding subtracting numbers in decimal form The digits to the right of the decimal point represent fractions with denominators that are powers of 0 For example, = When adding or subtracting numbers in decimal form, the decimal points must be lined up so that the digits in each column are of the same type For example to find : Sample Question g) Find h) Multiplying numbers in decimal form Numbers that have digits to the right of the decimal point are multiplied in the same way that whole numbers are multiplied except that it must be determined where the decimal point is placed in the product For example, since 5 6 = 4400, it follows that 5 6 = 5 6 = 5 6 = since multiplying by 000 means moving the decimal point places to the left, 5 6 = 4400 = 44 Sample Question h) Find 6 5 Basic Algebra a) Order of operations Expressions within parentheses, ( ), or brackets, [ ], are calculated first To reduce the number of parentheses brackets, it is understood that roots powers (exponents) are computed before multiplications divisions which in turn are calculated before additions subtractions For example, x means that is multiplied by the square of x ie if x = 4 then

6 x = 4 = 6 = 48 x does not mean: multiply by x then square the result ie if x = 4 then x ( 4) = = 44 Also for example, x + y means that the value of x is added to the square of y If x = 4 y = then x + y = 4 + = 4 + However, ( x + y) means the square of the sum of x + y:: if x = 4 y = then (x + y) = (4 + ) = 7 = 49 Sample Question a) Find the value of (5 + x )y + when x = y = 4 b) Rules for exponents If the base, b, is a positive constant, for example b =, then the following properties hold: p ) If p is a positive integer then b equals the result of multiplying b by itself p times For example, 4 = p q p+ q ) b b = b 5 for all real numbers p q For example = (because = ( ) ( ) π π+ ) = (The dot notation a b = a b is commonly used to prevent confusing the multiplication symbol with the variable x ) b p > 0 for all real numbers p p b p q = b 5 = q 4) b for all real numbers p q For example which we know is 5 p 4 b = = 7 = 9 q true because Note that because of property, b exists (or is q defined ) because b never equals 0 0 5) b = This follows from property 4) where p = q p b = p 6) b This follows from the previous three properties For example: = = = = b = 8 (ie if b 0, then b is the reciprocal of b (ie the number you multiply b with to get )

7 q p pq 6 6 7) ( b ) = b For example, ( ) = = ( ) = 4 = 64 = 8) If n is a positive integer greater than (ie n = or or 4 or ) / n then b is the nth root of b : the unique positive real number which when taken to the : / : nth power is b (note that b > 0) For example, : = : : = : because by property 6), : / : : : (: ) = : = : : / : which means that : = : because is the only positive real number which raise to the rd power makes 8 Sample Question b) : : : b (b ) : Express b as a single power of b c) Simplifying radicals Using the property: ab = a b, the square root of an expression is simplified by factoring out the largest square factor For example, : : : : : : : : x y = : : : x y : xy = : : xy : xy Sample Question c) : : : : : x + : : x y Simplify d) Multiplying expressions collecting like terms Products of constants variables (ie monomials) are said to be like terms if the variables have the same powers For example, xy 8xy are like terms may be added together using the distributive property: xy + 8xy = ( + 8)xy = xy (this is referred to as collecting like terms ) The terms xy x y are not like terms may not be represented by a single term Multiplying sums requires the use of the distributive principle For example: (xy + x y) = (xy + x y)(xy + x y) = xy (xy + x y) + x y(xy + x y) = 4 x y + 6x y + 6x y + 9x y = 4x y + x y + 9x y Sample question d) Exp collect like terms: (x + x + )(x x ) e) Factoring expressions Factoring a polynomial in one or more variables is the inverse of the previous topic You should know the following:

8 ) Factoring out the largest common factor For example: 5 x y + 7xy + 5xy = xy (x y ) ) Factoring the difference of squares: a b (a b)(a + b) For example: 9x 6y = (x) (4y) = (x 4y)(x + 4y) ) Factoring a perfect square trinomial: a ± ab + b = (a ± b) For example: 9x 4xy + 6y = (x) (x)(4y) + (4y) = (x 4y) 4) Factoring the difference of cubes: a b = (a b)(a + ab + b ) For example 8x 7y = (x) (y) = (x y)(4x + 6xy + 9y ) a + b = (a + b)(a ab + b ) 5) Factoring the sum of cubes: 6) Separating into groups: For example: x xy x + y = x(x y ) (x y ) = (x )(x y ) = (x )(x y)(x + y) Sample Question e) 4 Completely factor x y 7xy f) Simplifying rational expressions Using the rules for exponents (section b) a rational expression may be simplified For example, to simplify the following rational expression 4 (x y ) (xy ) 6x y so that the answer has no negative exponents: (9x y )( x y ) x y (x y ) (xy ) 4 4 = = = 4 6x y 6x y 6x y 8xy Sample Question f) Simplify express without negative exponents 5 4 (4x y ) (x y ) 4 (8x y ) Lines linear functions The word line means straight line The slope of a line in the xy-plane that contains the points ( x, y) ( x, y) is defined to be the y y x number x if x x is undefined if x = x If x = x ( y y

9 ) then the two points lie on a vertical line which has an undefined slope If x x y = y then the points lie on a horizontal line which has a slope of 0 Sample Question a) Find the value of b given that the slope of the line containing the points ( 5, b) is 7 (, ) b) The slope-intercept equation of a line is an equation of the form y = mx+c where m is the slope of the line (0,c) is the y-intercept of the line: the point where the line crosses the y-axis A solution of the equation y = mx + c is an ordered pair of numbers such that when x is substituted with the first number of the pair y is substituted with the second number of the pair, the equation is true For example, if the equation of a line is y = x+ then (4,) is a solution because = 4 + An ordered pair of numbers is a point on the line in the xy-plane if only if it is a solution of the equation of the line For example, if we are given that two points of a line are (,4) (,8) then the slope of 8 4 = this line is which means that the slope-intercept equation of the line is of the form y = x + c for some constant c Since (,4) is a point on the line, (,4) is a solution of y = x + c which means that 4 = + c c = Therefore the slope-intercept equation of the line is

10 y = x + Sample Question b) Find the slope-intercept equation of the line that contains (,8) (5,4) c) Parallel lines perpendicular lines Parallel lines do not cross each therefore have the same slope Lines are perpendicular to each other if they cross each other at right angles Theorem: Two non-vertical lines are perpendicular to each other if only if the product of their slopes equals For example if a line L is parallel to the line y = x+ contains the point (,9) then the equation of L is of the form y = x+c because its slope is Since (,9) is a solution: 9 = + c c =, the equation of L is y = x+ The line J that contains (,9) is perpendicular to the line y = x+ has an y = x + c = equation of the form because its slope is (because ) 9 = + c c = since (,9) is a solution: Therefore the equation of J is y = x + Sample Question c) Find the slope-intercept equations of the two lines that contain (4,8) such that one is parallel to, the other is perpendicular to the line y = 4x + 6 The intersection of two non-parallel lines is the point where they cross each other Since this point is a common solution to the equations of these lines, finding the point of intersection is equivalent to finding the solution of a system of two linear equations For example the intersection of the two lines (given by equations in the general form) x + y = 8 x 5y = is the solution of the system: : x + : y = : x 5y = One way to solve this system is to rewrite the first equation in slope-intercept form (ie solve for y in terms of x) then substitute into the second equation to get a equation entirely in terms of x Another way is to add multiples of the equations to eliminate a variable: for example, multiplying the first equation by the second equation by > yields the equivalent system: 6 x + 4y = 6 6 x + 5y = so that when the resulting equations are added we have 9y = 9, y = then by substituting into either equation we have x = so that the common solution (ie the d)

11 intersection of the lines) is (,) Sample Question d) Find the intersection of the lines 5x + 4y = x y = 5 e) The inverse of a linear function f(x) = mx+c If we write the f(x) = mx+c in the form y=mx+c (ie y = f(x)) then interchange x y to get the equation x = my+c, the variable y now represents the inverse function of f(x): f (x), so that solving for y gives the formula for f (x) For example, if f(x) = x+4, then writing y = x+4 y = x interchanging the x y gives x = y+4 or y = x-4 or which means that f (x) = x the inverse function Sample Question e) Find the inverse of f(x) = x 9 4 Parabolas quadratic functions A quadratic function (second degree polynomial) is a function of the form y = ax + bx + c where the coefficients a, b c are real numbers (constants) a 0 The graph of a quadratic function is a parabola If a >0 then the parabola opens up its vertex is its lowest point if a < 0 then the parabola opens down its vertex is the highest point Given any equation in x y, i the variable is replaced with the expression x d where d is some positive constant, then the graph of the new equation is the same as the graph of the original equation translated (or shifted) d units to the right (Replacing x with x + d d>0 results in a translation d to the left) For example, the graph of y = x is a parabola that has its vertex at (0,0) opens up The graph of y = x is formed by translating the points unit down so that the vertex is (0,-)

12 The graph of y = x right down x = (x ) is the graph of y = x translated to the a) The Quadratic formula: If ax + bx + c = 0 a 0 then b ± b 4ac x = a The quadratic formula can be used to find the x-intercepts of the parabola y = ax + bx + c If the discriminant b 4ac is a positive number, then the equation ax + bx + c = 0 has two real solutions, which means that the parabola y = ax + bx + c has two x-intercepts if the discriminant b 4ac equals 0, the parabola has only one x-intercept if the discriminant is a negative number, then the parabola has no x-intercepts which means that it lies entirely above or entirely below the x-axis

13 Sample Question 4a) Find the x-intercepts of the parabola y = x + 5x + b) Completing the square Rewriting a quadratic function y = ax + bx + c in the form y = a(x h) + k is known as completing the square The vertex of the parabola is (h,k) if a < 0 then k is the maximum value of y if a > 0 the k is the minimum value of y For example, rewriting y = x + x + 4 as y = (x + ) 8 means that the minimum value of y is 8 when x = - (because (x + ) 0 for all x (x + ) = 0 only if x = ) Therefore (, 8) is the lowest point on the parabola y = x + x + 4 therefore is the vertex To compete the square of an expression of the form x + bx, use the identity b b x + bx = (x ) 4 to complete the square of an expression of the form ax + bx, first factor out the coefficient a then use the previous identity For x + 5x + = (x + x) + = (x + ) + example: = (x + ) + = (x + ) Sample Question 4b) Find the maximum value of the function y = 4x + 8x + 0 c) Finding the equation y = ax + bx + c of a parabola given its vertex one other point Since the equation of such a parabola can be expressed in the form y = a(x h) + k, if the vertex (h,k) is known, only one other point is required to determine the value of a For example, if the vertex is (,4) another point is ( 4, 94), then the equation is of the form y = a(x + ) + 4 since (4,-94) is a solution, 94 = a(4 + ) + 4 = 49a + 4 a = Therefore the equation of the parabola is

14 y = (x + ) + 4 which written in stard form is y = x x 4 Sample Question 4c) Find the equation of the parabola with vertex (,) that contains the point (5,8) The equation must be in the form y = ax + bx + c 5 Logarithms Given a base b >, the logarithm of base b, log b (x) is the inverse of x the exponential function b This means that y = logb(x) if only if b y = x Since b y > 0 for all real y, it follows that log b x is defined only for x > 0 Special values are: log b () = 0 (because b 0 = ) log b (b) = (because b = b ) If the base b = 0, the logarithm is called the common logarithm is written as log(x), if the base is the constant e, the logarithm is called the natural logarithm is written ln(x) Properties of logarithms: logb (x) + logb (y) = logb (xy) for all positive real numbers x y x logb (x) logb (y) = logb y for all positive real numbers x y y logb (x ) = ylogb (x) for all positive real numbers x any real numbers y log (x) log (x) b a = log (a) 4 (Change of base) If a is another base, a >, then b Sample Question 5a) Find the value of log5 (5) log5() + log5(5) Sample Question 5b) logb (x) + logb (y) logb(w) Express as a single logarithm Sample Question 5c) 5 = Express 5 in logarithmic form Sample Question 5d) log x y If 5 = log (x ), express 5 in terms of y

15 6 Equations a) Linear equations in one variable For example, solve Exp the left side to get: 7 x = 6 x 0x + 0 = x, 6x = 7, 6 (x ) 5(4x 6) = x Sample Question 6a) Solve for x: (x 5) + 5(x 7) = 8(4x + ) b) Quadratic equations in one variable Rewrite the equation in the stard form then solve by using the Quadratic formula or by factoring For example: Solve (x + )(x ) = x + 6x x = x + 4x x = 0 (4x + )(x ) = 0 x = 4 or x = Sample Question 6b) Solve for x: ( x 0)(x + ) = (4x 9)(x + 6) c) Equations involving rational expressions are sometimes disguised quadratic x + = x + 6 equations eg Solve x Multiplying both sides by x : x + = (x + 6)(x ), x + = x + 5x 6, x + x 8 = 0, ( x + 4)(x ) x = 4 or x = Sample Question 6c) x + 4 Solve for x: x = x d) Equations involving absolute values must be considered in separate cases For x = x + example, to solve the two cases to consider are: Case : If x, then x = x x = x + the equation becomes, x 9 = x +,x = 0, x = 5 which is a solution because

16 5 > x = (x ) = x Case ): If x < then, the equation becomes x = x +, 9 x = x +, 4x = 8, x = which is a solution since < Therefore the equation has two solutions: x = x = 5 Sample Question 6d) 4 x = x + Find all the solutions of e) Equations involving radicals If the equation contains one radical, the radical must first be isolated on one side of the equation then when both sides of the equation are squared, the resulting equation will have no radicals, but may have more solutions than the original equation The solutions of the new equation must be checked to see that they are also solutions of the given equation For example: To solve x + x + 4 = 5x 7, we first isolate the radical: x + 4 = x 7 then square both sides: x + 4 = 4x 8x + 49, 9 9 4x ( 4x 9)(x 5) = 0 x = x = 9x + 45 = 0, 4 or x = 5 When 4, the left x + x + 4 = + = side of the given equation, 4 4 but the right side of the given x 7 = = equation, Therefore x = 4 is not a solution of the given equation ( is called an extraneous solution) Since x=5 makes both sides of the given equation equal to 8, the given equation has only one solution: x = 5 Sample Question 6e) Find all the solutions of 4 x + x + = 5 f) Equations involving logarithms Solving such equations requires knowing the definition of log b (x) the properties of logarithms Extraneous solutions may occur For example: Solve log (x + ) + log(x + ) = 7 : log (x + ) + log(x + ) = 7 log(x + 0x + ) = log(8) x + 0x + = 8, x + 0x 5 = 0, 5 ( x + 5)(x 5) = 0 x = or x = 5 However, only x = 5 is a solution of the

17 5 x = given equation because when, 6 log(x + ) = log is undefined Sample Question 6f) Solve log (x + ) log(x) = g) Exponential equations To use the rules for exponents, the equation should be rewritten so that there is only one base For example: 4x x+ x+ 9 = To solve: we use the common base : x+ x+ 4x x+ 6x+ 4 4x 7x+ 5 4 ( ) = ( ) = = 7x + 5 = 4 9 x = (because exponential functions are one-to-one), 7x = 9, 4 Sample Question 6g) x x = Solve 64 x+ 7 Inequalities a) Linear inequalities in one variable Solving a linear inequality is similar to solving a linear equation except that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality must be reversed For example, to solve the linear inequality (x + 5) > 4(5x 8) : (x + 5) > 4(5x 8) 6x + 5 > 0x 4x > 47 after dividing 47 x < both sides by 4 we have 4 so that the solution of the inequality is the set { x : x < } (, ) 4 which may also be given using interval notation: 4 Sample Question 7a) Find the set of all solutions of: (x ) > 4x + 5 b) The graph of the solutions of a linear inequality in two variables x y The solutions of the inequality y > mx + c is the set of points that lie above the line y = mx + c (which is indicated with a dotted line) The solutions of the inequality y mx + c is the set of points that are on or above the line y = mx + c (which is indicated by a solid line)

18 The solutions of y < mx + c y mx + c are the sets of points under or under on the line y = mx + c To draw the graph a linear inequality in x y, we first rewrite it in one of the four forms For example, to find the graph of the inequality y x, we rewrite the inequality as y x + y x + y = x +, then draw the graph of the line shade the area above the line: Sample Question 7b) Graph the set of solutions of x y 6 c) Inequalities involving polynomials of one variable To solve an inequality of the form P(x) > 0 (or P(x) < 0 or P(x) 0 or P(x) 0 ) where P(x) is a polynomial of degree >, we first find the solutions of P(x) = 0 then determine the sign of P(x) over each of the intervals formed by these solutions For example: to solve x x x we rewrite as x (x ) (x ), x (x ) (x ) 0, ( x )(x ) 0 note that the equation ( x )(x ) = 0 has three solutions: x =, x = x = We can use a sign chart to determine the intervals where values the intervals where it has negative values : ( x )(x ) has positive The sign of ( x )(x ) over each interval can be determined by using a test number or by analyzing the factors The solution set for the given question is { x : x or x } Using interval notation, the solution set is

19 [,] [, ) Sample Question 7c) Find all the solutions of x + x 6x d) Inequalities in one variable that involve absolute values If c > 0 then x < c c < x < c x > c x > c or x < c For example, x + < 9 9 < x + < 9 0 < x < 8 5 < x < 4 5 x 4 5 x 4 or 5 x 4 x or x 9 x or x Sample Question 7d) Find the solution set of 4x Trigonometry You need to know the definitions of cos(θ ), sin(θ) tan(θ) for angles between 0 60, the exact values of cos(θ ), sin(θ) tan(θ) of 0, 45, 60 how to compute the exact values of cos(θ ), sin(θ) tan(θ) of any integer multiple of any of these angles You need to know the identity: cos θ + sin θ = Sample Question 8a) Find the exact value of tan(40 ) Sample Question 8b) Given that 90 < A < 80 that sin(a) = 5, find the exact value of cos(a) Sample Question 8c) Find the exact the diagram above length of side x in

20 9 Geometry a) The areas of rectangles triangles The area of a rectangle of width x length y bh is xy The area of a triangle of base b height h is Sample Question 9a) Find the exact area of the triangle shown below: b) The area of a circle of radius r is π r πr the circumference (distance around) is Sample Question 9b) If the area of a circle is 5 π, what is its circumference? c) Similar triangles If a pair of triangles have the same angles, then they are said to similar triangles the ratios of their corresponding sides are equal

21 Sample Question 9c) In the above diagram, AD = 8, DB = 4, DE = 6 DE is parallel to BC Find the length of BC Answers a) 0 b) 59 c) 5 h) d) 5 5 e) 4 8 f) g) a) 95 b) b 4 5 x + y 4 c) d) 6x + 5x + x 8x 0 e) xy (x 9y ) = xy (x y)(x + y) f) 6x y 9 b ( ) = 7 = 5 y = x + a) 5 b) c) The parallel line is y = 4x 8 y = x + 9 the perpendicular line is 4 d) Adding two times the second equation to the first gives x =, x = from

22 y = (, ) either equation, Therefore the solution is e) Setting y = x 9 where y represents f(x) then interchanging x y to get f (x) = x + x = y 9 where y now represents the inverse of f, y = x ± ± 4 x = = 4a) Using the Quadratic formula, y = 0 if 4 4 Therefore ,0 the x-intercepts are 4,0 4 y = 4x + 8x + 0 = 4(x x) + 0 = 4 (x ) + 4b) ( ) 0 = 4(x ) + 4 Since 4(x ) 0 for all x, the maximum value of y is 4 4c) Since the vertex is (,), the equation of the parabola is of the form y = a(x ) + since (5,8) is a solution of this equation: 8 = 4a + a = 4 Therefore the equation of the parabola is which in stard form is y = 4x 4x + 8 y = 4(x ) + 5a) log5 (5) log5() + log5(5) = 5 log5 + log5(5 ) = log5(5) + log5(5) = 4 logb (x) + logb (y) logb(w) 5b) = logb (x) + logb (y ) logb( xy = log b w 5 = log 5 = 5c) 5 5 w ) 5d) y y y log x y 5 x 5 x 5 5 = = = = x log5(x ) = y

23 6a) (x 5) + 5(x 7) = 8(4x + ) 6x 0 + 0x 5 = x x = 7 x = 48 6b) ( x 0)(x + ) = (4x 9)(x + 6) 6x 7x 0 = 4x + 5x 54 6 ± x = x x + 44 = 0 x 6x + = 0 x = 8 ± 4 x + 4 = x 6c) x Multiplying both sides by 5x+0 gives: 0x + 0 (x + 6) = (x )(5x + 0) 9x + 4 = 5x + 5x x x = or 4 + 6x 04 = 0 (5x + 6)(x 4) = 0 5 after checking, both values are seen to be solutions of the given equation 4 x = x + 6d) 4 x 4 = x + 6x = x + 4 4x = 6 x = 4 Case : x : Since 4 >, x= 4 is a solution of the given equation 4 4 x = x + 6x = x + 4 8x = 8x = 8 x = Case : x < : Since <, x= is also a solution of the given equation Therefore the given equation has two solutions: x= x = 4 6e) 4 x + x + = 5 After subtracting 4x from both sides then squaring both sides: x + = (5 4x) x + = 6x 0x + 5 6x x + = 0 8x 6x + = 0 (8x 7)(x ) = 0 x = 7 8 solution of the given equation x + log = log() log (x ) log (x) 6f) + = x or but only x = is a

24 x + = x + = x x = x which is indeed a solution of the given equation x+ x x = 6g) 64 Choosing 4 as a common base: x 4 x+ 9 x 5x 5 x 4 4 = = 4 5x + 5 = x 8x = 8 x = 5 (x ) > 4x + 5 x > 4x + 5 x > 6 x < 7a) 5 7b) x y 6 y x + 6 y x Therefore the solution set is the y = x set of points that are on or above the line : 7c) x + x 6x : First find the solutions of the equality x + x 6x = 0 x(x + x 6) = 0 x(x )(x + ) = 0 x = 0 or or Therefore the solution set is { x : x or 0 x } (or in interval notation: (, ] [0,] ) 4x 8 7d) + 4x + 8 or 4x + 8 4x 6 or 4x 0 x or 5 5 x { x : x or x } ie the solution set is (in interval

25 5 (, ] [, ) notation: ) 8a) o o cos(40 ) = sin(40 ) = Using the reference angle 60,, so that tan(40 o ) = 9 4 cos(a) = sin (A) = = 8b) Since A is in the second quadrant, 5 5 8c) x = o 0cos(0 ) = 5 9a) h = o 0sin(0 ) = 5 the base b is 8 Therefore the area of the triangle is bh = 45 9b) Area = π r = 5π r = 5 Therefore the circumference is πr = 5π BC BA BC = = BC = 9 9c) Since ADE is similar to ABC: DE AD 6 8