MAIDSTONE GRAMMAR SCHOOL FOR GIRLS

Size: px
Start display at page:

Download "MAIDSTONE GRAMMAR SCHOOL FOR GIRLS"

Transcription

1 MAIDSTONE GRAMMAR SCHOOL FOR GIRLS King Edward VI High School DEPARTMENT OF MATHEMATICS Introduction to A level Maths INDUCTION BOOKLET

2 CONTENTS Reading List... 3 Section 1: FRACTIONS... 4 Section : EXPANDING... 6 Section 3: LINEAR EQUATIONS... 8 Section 4: LINEAR INEQUALITIES Section 5: SIMULTANEOUS EQUATIONS Section 7: SOLVING QUADRATIC EQUATIONS Section 8: CHANGING THE SUBJECT OF A FORMULA... 0 Section 9: INDICES... 3 Section 10: SURDS... 6 Section 11: FUNCTIONS Practice Booklet Test Solutions to the Exercises... 3 Solutions to the Practice Booklet Test READING LIST As a student who is choosing to study Mathematics at A Level, it is logical to assume that you have an interest in the subject. With that said, the following books may be of interest to you. Alex s Adventures in Numberland by Alex Bellos Cabinet of Mathematical Curiosities by Ian Stewart The Num8er My5teries by Marcus du Sautoy How Many Socks Make a Pair?: Surprisingly Interesting Maths by Rob Eastway The Curious Incident of the Dog in the Night-time by Mark Haddon The Penguin Dictionary of Curious & Interesting Numbers by David Wells The Calculus Wars by Jason Socrates Bardi The Code Book by Simon Singh 50 Mathematical Ideas You Really Need to Know by Tony Crilly

3 Section 1: FRACTIONS To add or subtract fractions, find the lowest common denominator of the two fractions and then rewrite the fractions accordingly. Ensure that you simplify as far as possible. Examples When multiplying fractions, it is far more efficient to cancel first; this avoids trying to simplify fractions with unnecessarily large numerators and/or denominators. To multiply with fractions, simply multiply the numerators and denominators together. Example To divide by a fraction, we simply multiply by the reciprocal of the second fraction (i.e. we flip the second fraction over ). Example

4 For addition and subtraction with mixed numbers, add (or subtract) the integer (whole number) parts first and then work with the fractions. Examples To multiply and divide with mixed, convert the mixed numbers to improper fractions and then calculate as normal. It should also be noted that in the study of A Level Mathematics, answers are preferred as improper fractions rather than mixed numbers. EXERCISE More help is available from MyMaths: Adding/Subtracting Fractions, Multiplying Fractions, Dividing Fractions, Mixed Numbers

5 Section : EXPANDING To remove a single bracket multiply every term in the bracket by the number or expression outside: Examples 1) 3 (x + y) = 3x + 6y ) -(x - 3) = (-)(x) + (-)(-3) = -4x + 6 To expand two brackets multiply everything in the first bracket by everything in the second bracket. You may have used * the smiley face method * FOIL (First Outside Inside Last) * using a grid. Examples: 1) (x + 1)(x + ) = x(x + ) + 1(x + ) or (x +1)(x + ) = x + + x + x = x + 3x + or x 1 x x x x (x +1)(x + ) = x + x + x + = x + 3x + ) (x - )(x + 3) = x(x + 3) - (x +3) = x + 3x 4x - 6 = x x 6 or (x - )(x + 3) = x 6 + 3x 4x = x x 6 or x - x x -4x 3 3x -6 (x +3)(x - ) = x + 3x - 4x - 6 = x - x - 6

6 EXERCISE A 1. 7(4x + 5). -3(5x - 7) 3. 5a 4(3a - 1) 4. 4y + y( + 3y) 5. -3x (x + 4) Multiply out the following brackets and simplify. 7. (x + )(x + 3) 8. (t - 5)(t - ) 9. (x + 3y)(3x 4y) 10. 4(x - )(x + 3) 11. (y - 1)(y + 1) 6. 5(x - 1) (3x - 4) 1. (3 + 5x)(4 x) Two Special Cases Perfect Square: Difference of two squares: (x + (a) = (x + (a)(x + (a) = x + ax + a (x - (a)(x + (a) = x a (x - 3) = (x 3)(x 3) = 4x 1x + 9 (x - 3)(x + 3) = x 3 = x 9 EXERCISE B Expand the following 1. (x - 1). (3x + 5) 3. (7x - ) 4. (x + )(x - ) 5. (3x + 1)(3x - 1) 6. (5y - 3)(5y + 3 More help is available from MyMaths: Brackets

7 Section 3: LINEAR EQUATIONS When solving an equation whatever you do to one side must also be done to the other. You may add the same amount to both side subtract the same amount from each side multiply the whole of each side by the same amount divide the whole of each side by the same amount. If the equation has unknowns on both sides, collect all the letters onto the same side of the equation. If the equation contains brackets, you often start by expanding the brackets. A linear equation contains only numbers and terms in x. (Not x or 3 x or 1/x et(c) Example 1: Solve the equation 64 3x = 5 Solution: There are various ways to solve this equation. One approach is as follows: Step 1: Add 3x to both sides (so that the x term is positiv(e): 64 = 3x + 5 Step : Subtract 5 from both sides: Step 3: Divide both sides by 3: 39 = 3x 13 = x So the solution is x = 13. Example : Solve the equation 6x + 7 = 5 x. Solution: Step 1: Begin by adding x to both sides 8x + 7 = 5 (to ensure that the x terms are together on the same sid(e) Step : Subtract 7 from each side: 8x = - Step 3: Divide each side by 8: x = -¼ Exercise A: Solve the following equations, showing each step in your working: 1) x + 5 = 19 ) 5x = 13 3) 11 4x = 5 4) 5 7x = -9 5) x = 8 x 6) 7x + = 4x 5

8 Example 3: Solve the equation (3x ) = 0 3(x + ) Step 1: Multiply out the brackets: 6x 4 = 0 3x 6 (taking care of the negative signs) Step : Simplify the right hand side: 6x 4 = 14 3x Step 3: Add 3x to each side: 9x 4 = 14 Step 4: Add 4: 9x = 18 Step 5: Divide by 9: x = Exercise B: Solve the following equations. 1) 5(x 4) = 4 ) 4( x) = 3(x 9) 3) 8 (x + 3) = 4 4) 14 3(x + 3) =

9 EQUATIONS CONTAINING FRACTIONS When an equation contains a fraction, the first step is usually to multiply through by the denominator of the fraction. This ensures that there are no fractions in the equation. y Example 4: Solve the equation 5 11 Solution: Step 1: Multiply through by (the denominator in the fraction): y 10 Step : Subtract 10: y = 1 Example 5: Solve the equation 1 (x 1) 5 3 Solution: Step 1: Multiply by 3 (to remove the fraction) x 1 15 Step : Subtract 1 from each side x = 14 Step 3: Divide by x = 7 When an equation contains two fractions, you need to multiply by the lowest common denominator. This will then remove both fractions. Example 6: Solve the equation x 1 x 4 5 Solution: Step 1: Find the lowest common denominator: The smallest number that both 4 and 5 divide into is 0. Step : Multiply both sides by the lowest common denominator Step 3: Simplify the left hand side: 0( x 1) 0( x ) ( x 1) 0 ( x ) (x + 1) + 4(x + ) = 40 Step 4: Multiply out the brackets: 5x x + 8 = 40 Step 5: Simplify the equation: 9x + 13 = 40 Step 6: Subtract 13 9x = 7 Step 7: Divide by 9: x = 3

10 Example 7: Solve the equation x 3 5x x 4 6 Solution: The lowest number that 4 and 6 go into is 1. So we multiply every term by 1: 1( x ) 1(3 5 x) 1x Simplify 1x 3( x ) 4 (3 5 x) Expand brackets 1x 3x x Simplify 15x x Subtract 10x 5x 6 18 Add 6 5x = 4 Divide by 5 x = 4.8 Exercise C: Solve these equations 1) 1 ( 3) 5 x ) x x ) y y x 3 x 3 5 4) ) 7x 1 13 x y 1 y 1 y 5 6) 3 6 7) x 1 5x 3 x 8) x x

11 FORMING EQUATIONS Example 8: Find three consecutive numbers so that their sum is 96. Solution: Let the first number be n, then the second is n + 1 and the third is n +. Therefore n + (n + 1) + (n + ) = 96 3n + 3 = 96 3n = 93 n = 31 So the numbers are 31, 3 and 33. Exercise D: 1) Find 3 consecutive even numbers so that their sum is 108. ) The perimeter of a rectangle is 79 cm. One side is three times the length of the other. Form an equation and hence find the length of each side. 3) Two girls have 7 photographs of celebrities between them. One gives 11 to the other and finds that she now has half the number her friend has. Form an equation, letting n be the number of photographs one girl had at the beginning. Hence find how many each has now. More help is available from MyMaths: Solving Equations

12 Section 4: LINEAR INEQUALITIES Linear inequalities can be solved using the same techniques as linear equations (for the most part). We may add and subtract the same numbers on both sides and we can also multiply and divide by positive numbers; multiplying/dividing both sides by a negative needs further explanation. Example x - 3 < 11 Here we can simply add 3 to both sides: x < 14 Next, as with linear equations we divide by : x < 7 However, if we were to have 3 x > 6, we would need to adopt a different technique. If we wish to divide or multiply by a negative number, we must reverse the direction of the inequality. Example As before, we would subtract 3 from both sides: Divide by - and subsequently reverse the inequality: We can see this working on a more basic level; it is true to state that 3 < 4 but it is incorrect if we multiply both sides by a negative and keep the sign as it was: -6 < -8 is not true. You may find it easier to rearrange the inequality: Example If we add x to both sides, we remove the hassle: We then subtract 6: Divide by two as normal: Remember that you can change this round to say Both of these techniques are acceptable and is more a matter of preference. Exercise: Solve each inequality 1) ) 3) 4) 5) 6) 7) 8) More help is available from MyMaths: Inequalities, Negative Inequalities

13 Section 5: SIMULTANEOUS EQUATIONS Example 3x + y = 8 5x + y = 11 x and y stand for two numbers. Solve these equations in order to find the values of x and y by eliminating one of the letters from the equations. In these equations it is simplest to eliminate y. Make the coefficients of y the same in both equations. To do this multiply equation by, so that both equations contain y: 3x + y = 8 10x + y = = To eliminate the y terms, subtract equation from equation. We get: 7x = 14 i.e. x = To find y substitute x = into one of the original equations. For example put it into : 10 + y = 11 y = 1 Therefore the solution is x =, y = 1. Remember: Check your solutions by substituting both x and y into the original equations. Example: Solve x + 5y = 16 3x 4y = 1 Solution: Begin by getting the same number of x or y appearing in both equation. Multiply the top equation by 4 and the bottom equation by 5 to get 0y in both equations: 8x + 0y = 64 15x 0y = 5 As the SIGNS in front of 0y are DIFFERENT, eliminate the y terms from the equations by ADDING: 3x = 69 + i.e. x = 3 Substituting this into equation gives: 6 + 5y = 16 5y = 10 So y = The solution is x = 3, y =. Exercise: Solve the pairs of simultaneous equations in the following questions: 1) x + y = 7 ) x + 3y = 0 3x + y = 9 3x + y = -7 3) 3x y = 4 4) 9x y = 5 x + 3y = -6 4x 5y = 7 5) 4a + 3b = 6) 3p + 3q = 15 5a 4b = 43 p + 5q = 14 More help is available from MyMaths: Simultaneous Equations 1, Simultaneous Equations - Medium, Simultaneous Equations - Hard, Simultaneous Equations - Negatives, Solving Simultaneous Equations Graphically

14 Section 6: FACTORISING Taking out a common factor Example 1: Factorise 1x 30 Solution: Example : 6 is a common factor to both 1 and 30. Factorise by taking 6 outside a bracket: 1x 30 = 6(x 5) Factorise 6x xy Solution: is a common factor to both 6 and. Both terms also contain an x. Factorise by taking x outside a bracket. 6x xy = x(3x y) Example 3: Factorise 9x 3 y 18x y Solution: 9 is a common factor to both 9 and 18. The highest power of x that is present in both expressions is x. There is also a y present in both parts. So we factorise by taking 9x y outside a bracket: 9x 3 y 18x y = 9x y(xy ) Example 4: Factorise 3x(x 1) 4(x 1) Solution: There is a common bracket as a factor. So we factorise by taking (x 1) out as a factor. The expression factorises to (x 1)(3x 4) Exercise A Factorise each of the following 1) 3x + xy ) 4x xy 3) pq p q 4) 3pq - 9q 5) x 3 6x 6) 8a 5 b 1a 3 b 4 7) 5y(y 1) + 3(y 1) More help is available from MyMaths: Factorising Expressions

15 Factorising quadratics Simple quadratics: Factorising quadratics of the form x bx c The method is: Step 1: Form two brackets (x )(x ) Step : Find two numbers that multiply to give c and add to make b. Write these two numbers at the end of the brackets. Example 1: Factorise x 9x 10. Solution: Find two numbers that multiply to make -10 and add to make -9. These numbers are -10 and 1. Therefore x 9x 10 = (x 10)(x + 1). General quadratics: Factorising quadratics of the form ax bx c One method is that of combining factors. Look at factorising on MyMaths or ask a teacher for help with our preferred method but is difficult to explain on paper. Another method is: Step 1: Find two numbers that multiply together to make ac and add to make b. Step : Split up the bx term using the numbers found in step 1. Step 3: Factorise the front and back pair of expressions as fully as possible. Step 4: There should be a common bracket. Take this out as a common factor. Example : Factorise 6x + x 1. Solution: We need to find two numbers that multiply to make 6-1 = -7 and add to make 1. These two numbers are -8 and 9. Therefore, 6x + x 1 = 6x - 8x + 9x 1 = x(3x 4) + 3(3x 4) (the two brackets must be identical) = (3x 4)(x + 3) Difference of two squares: Factorising quadratics of the form x a Remember that x a = (x + (a)(x (a). Therefore: x x x x 9 3 ( 3)( 3) 16x 5 ( x) 5 (x 5)(x 5) Also notice that: and x 8 ( x 4) ( x 4)( x 4) 3 3x 48xy 3 x( x 16 y ) 3 x( x 4 y)( x 4 y) Factorising by pairing or grouping Factorise expressions like x xy x y using the method of factorising by pairing: x xy x y = x(x + y) 1(x + y) (factorise front and back pairs, both brackets identical) = (x + y)(x 1)

16 Exercise B Factorise 1) x x 6 8) 10x 5x 30 ) x 6x 16 9) 4x 5 3) x 5x 10) x 3x xy 3y 4) x 3 x 11) 4x 1x 8 5) 3x 5x 1) 16m 81n 6) y 17y 1 13) 3 4y 9a y 7) 7y 10y 3 14) 8( x 1) ( x 1) 10 More help is available from MyMaths: Factorising Quadratics 1, Factorising Quadratics

17 Section 7: SOLVING QUADRATIC EQUATIONS A quadratic equation has the form ax bx c 0. There are two methods that are commonly used for solving quadratic equations: * factorising * the quadratic formula Not all quadratic equations can be solved by factorising. Method 1: Factorising Make sure that the equation is rearranged so that the right hand side is 0. It usually makes it easier if the coefficient of x is positive. Example 1 : Solve x 3x + = 0 Factorise (x 1)(x ) = 0 Either (x 1) = 0 or (x ) = 0 So the solutions are x = 1 or x = Note: The individual values x = 1 and x = are called the roots of the equation. Example : Solve x x = 0 Factorise: x(x ) = 0 Either x = 0 or (x ) = 0 So x = 0 or x = More help is available from MyMaths: Quadratic Equations

18 Method : Using the formula The roots of the quadratic equation ax bx c 0 are given by the formula: b b 4ac x a Example 3: Solve the equation x x Solution: First we rearrange so that the right hand side is 0. We get We can then tell that a =, b = 3 and c = -1. Substituting these into the quadratic formula gives: x 3x ( 1) x (this is the surd form for the solutions) 4 If we have a calculator, we can evaluate these roots to get: x = 1.81 or x = Exercise 1) Use factorisation to solve the following equations: (a) x + 3x + = 0 (b) x 3x 4 = 0 (c) x = 15 x ) Find the roots of the following equations: (a) x + 3x = 0 (b) x 4x = 0 (c) 4 x = 0 3) Solve the following equations either by factorising or by using the formula: (a) 6x - 5x 4 = 0 (b) 8x 4x + 10 = 0 4) Use the formula to solve the following equations to 3 significant figures where possible (a) x +7x +9 = 0 (b) 6 + 3x = 8x (c) 4x x 7 = 0 (d) x 3x + 18 = 0 (e) 3x + 4x + 4 = 0 f) 3x = 13x 16 More help is available from MyMaths: The Quadratic Formula

19 Section 8: CHANGING THE SUBJECT OF A FORMULA Rearranging a formula is similar to solving an equation always do the same to both sides. Example 1: Make x the subject of the formula y = 4x + 3. Solution: y = 4x + 3 Subtract 3 from both sides: y 3 = 4x Divide both sides by 4; y 3 x 4 y 3 So x is the same equation but with x the subject. 4 Example : Make x the subject of y = 5x Solution: Notice that in this formula the x term is negative. y = 5x Add 5x to both sides y + 5x = (the x term is now positiv(e) Subtract y from both sides 5x = y Divide both sides by 5 y x 5 Example 3: 5( F 3) The formula C is used to convert between Fahrenheit and Celsius. 9 Rearrange to make F the subject. 5( F 3) C 9 Multiply by 9 9C 5( F 3) (this removes the fraction) Expand the brackets 9C 5F 160 Add 160 to both sides 9C 160 5F Divide both sides by 5 9C 160 F 5 9C 160 Therefore the required rearrangement is F. 5 Exercise A Make x the subject of each of these formulae: 1) y = 7x 1 x 3) 4y 3 x 5 ) y 4(3x 5) 4 4) y 9

20 Example 4: Make x the subject of Solution: Subtract y from both sides: x y w x y w Square root both sides: x w y Remember the positive & negative square root. x w y (this isolates the term involving x) Example 5: Make a the subject of the formula t 1 5a 4 h Solution: Multiply by 4 Square both sides Multiply by h: Divide by 5: 1 5a t 4 h 5a 4t h 5a 16t h 16t h 5a 16th a 5 Exercise B: Make t the subject of each of the following 1) P wt 3r 3) V 1 3 t h 5) Pa w( v t) g ) wt P 3r 4) P t g 6) r a bt More help is available from MyMaths: Rearranging Equations

21 Harder examples Sometimes the subject occurs in more than one place in the formula. In these questions collect the terms involving this variable on one side of the equation, and put the other terms on the opposite side. Example 6: Make t the subject of the formula a xt b yt Solution: a xt b yt Start by collecting all the t terms on the right hand side: Add xt to both sides: a b yt xt Now put the terms without a t on the left hand side: Subtract b from both sides: a b yt xt Factorise the RHS: a b t( y x) Divide by (y + x): So the required equation is a b t y x t a b y x Example 7: Make W the subject of the formula Wa T W b Solution: This formula is complicated by the fractional term. Begin by removing the fraction: Multiply by b: bt bw Wa Add bw to both sides: bt Wa bw (this collects the W s together) Factorise the RHS: bt W ( a b) Divide both sides by a + b: W bt a b Exercise C Make x the subject of these formulae: 1) ax 3 bx c ) 3( x a) k( x ) 3) x 3 y 5x 4) x x 1 a b More help is available from MyMaths: Higher Rearranging

22 Section 9: INDICES Basic rules of indices y 4 means y y y y. 4 is called the index (plural: indices), power or exponent of y. There are 3 basic rules of indices: 1) ) 3) ( a ) m n m n a a a e.g. m n m n a a a e.g. m n mn Further examples a e.g y 5y 5y a 6a 4a (multiply the numbers and multiply the a s) c 3c 6 6c 8 (multiply the numbers and multiply the c s) 7 7 4d 5 4d 3d 8d (divide the numbers and divide the d terms by subtracting the powers) 3d Exercise A Simplify the following: Remember that b b 1 1) ) 3) 4) 5) b 5b 5 3c c 5 b c bc 3 6 n ( 6 n ) 8n n 8 3 6) d d ) a 4 8) 3 d

23 Zero index: Remember 0 a 1 For any non-zero number, a. 0 Therefore Negative powers 0 3 A power of -1 corresponds to the reciprocal of a number, i.e. Therefore This result can be extended to more general negative powers: This means: Fractional powers: Fractional powers correspond to roots: In general: Therefore: 1/ a 1/n n a 0 a 1 1 (Find the reciprocal of a fraction by turning it upside down) a n a 1. n a / 1/ 3 3 1/ 4 4 a a a a a a 1/ / m / n 1/ n A more general fractional power can be dealt with in the following way: a a So 3 3/ / 3 1/ / 3/ m

24 Exercise B: Find the value of: 1) ) 1/ 4 1/ 3 7 4) 5) ) 3 11) 8 7 / 3 3) 1 1/ 9 6) 7) 7 1 /3 7 9) 8 / 3 10) 1/ ) / Simplify each of the following: 13) 14) a 3a 3 x x 1/ 5 / 4 15) x y 1/ More help is available from MyMaths: Indices 1, Indices, Indices 3

25 Section 10: SURDS A surd is a root of a number that cannot be expressed as an integer. Surds are part of the set of irrational numbers. Example: and are surds but is not as it equals. Simplifying Surds Start to simplify surds by using two rules: and By using the multiplication rule, simplify surds by finding a factor of the number you are taking a root of which is a square number. Always try to find the largest square number factor, otherwise you will have to simplify further. Example: Exercise A Simplify 1) ) 3) 4) 5) 6)

26 Multiplying and Dividing with Surds The rules of algebra are true for any numeric value; these include surds. To multiply and divide using surds deal with any integers together and then deal with any surds. Examples: ( ) ( )( ) ( )( ) ( ) In this example, you could expand as usual but this is an example of the difference of two squares. Exercise B Simplify 1) ) 3) 4) 5) 6) 7) ( ) 8) ( )

27 Addition and Subtraction with Surds You can only add or subtract with surds if the surd is the same; sometimes if they are not the same, you may be able to simplify them so that the same surd is present. Example: Here add the and as the same surd is present but you cannot add the. By simplifying to, you can add the two surds together. These methods also work for subtraction of surds. Exercise C Simplify 1) ) 3) 4) 5) 6) 7) 8) 9) 10)

28 Rationalising the Denominator It is far easier to calculate with a fraction if the surd if the denominator is a rational number (i.e. not a sur(d). The process of this is known as rationalising the denominator. To do this, multiply by the surd in the denominator, doing so makes use of the fact that ( ) Example: Multiply the denominator by to rationalise it and so multiply the numerator by also: Example : Example 3: ( ) Exercise D Simplify 1) ) 3) 4) 5) 6) 7) 8) 9) More help is available from MyMaths: Surds 1, Surds (Surds covers some material which is not here; this will be covered in Core 1)

29 Section 11: FUNCTIONS A function is like a machine: it has an input and an output. The output is related somehow to the input. f(x) tells us the rule for finding the output for any given input. f(x) is one such function and is read as f of x. Other letters may be used for functions too. Example: If a) b) a) b), find If c) d) c) d), find Exercise A Given that and, find 1) ) 3) 4) 5) 6) Given the rule and an output generated, we can find the input. Example: If and, find. If and, find the possible values of. ` Exercise B 1. Given that and find. Given that and find the possible values of

30 Practice Booklet Test You may NOT use a calculator If ax b + bx + c = 0 then x = b 4ac a 1. Calculate (a) (b) (c) (d). Expand and simplify (a) (x + 3)(x 1) (b) (a + 3) (c) 4x(3x ) x(x + 5) 3. Factorise (a) x 7x (b) y 64 (c) x + 5x 3 (d) 6t 13t Simplify 3 4x y (a) 3 8x y (b) 3x + 3 4x Solve the following equations h 1 3h (a) + = 4 (b) x 8x = 0 (c) p + 4p = Write each of the following as single powers of x and / y 1 (a) (b) (x y) 3 5 x (c) 4 x x 7. Work out the values of the following, giving your answers as fractions (a) 4 - (b) (c) 7 8. Solve the simultaneous equations 3x 5y = -11 5x y = 7 9. Rearrange the following equations to make x the subject (a) v = u 1 + ax (b) V = πx x h (c) y = 3 x Solve 5x x 1 = 0 giving your solutions in surd form 11. Find the values of x which satisfy the following inequalities (a) (b) 1. Given and, find (a) (b) (c) Simplify (a) (b) (c) (d) (e)

31 Solutions to the Exercises SECTION 1 1) 4) 7) 10) ) 5) 8) 11) 3) 6) 9) 1) SECTION Ex A 1) 8x + 35 ) -15x + 1 3) -7a + 4 4) 6y + 3y 5) -4x 4 6) 7x 1 7) x + 5x + 6 8) t 3t 10 9) 6x + xy 1y 10) 4x + 4x 4 11) 4y 1 1) x 5x Ex B 1) x x + 1 ) 9x + 30x + 5 3) 49x 8x + 4 4) x 4 5) 9x -1 6) 5y 9 SECTION 3 Ex A 1) 7 ) 3 3) 1½ 4) 5) 6) Ex B 1).4 ) 5 3) 1 4) ½ Ex C 1) 7 ) 15 3) 4/7 4) 35/3 5) 3 6) 7) 9/5 8) 5 Ex D 1) 34, 36, 38 ) 9.875, ) 4, 48 SECTION 4 1) ) 3) 4) 5) 6) 7) 8) SECTION 5 1) x = 1, y = 3 ) x = -3, y = 1 3) x = 0, y = - 4) x = 3, y = 1 5) a = 7, b = - 6) p = 11/3, q = 4/3

32 SECTION 6 Ex A 1) x(3 + y) ) x(x y) 3) pq(q p) Ex B 1) (x 3)(x + ) ) (x + 8)(x ) 3) (x + 1)(x + ) 4) x(x 3) 5) (3x -1 )(x + ) 4) 3q(p 3q) 5) x (x - 3) 6) 4a 3 b (a 3b ) 6) (y + 3)(y + 7) 7) (7y 3)(y 1) 8) 5(x 3)(x + ) 9) (x + 5)(x 5) 10) (x 3)(x y) 7) (y 1)(5y + 3) 11) 4(x )(x 1) 1) (4m 9n)(4m + 9n) 13) y(y 3(a)(y + 3(a) 14) (4x + 5)(x 4) SECTION 7 1) (a) -1, - (b) -1, 4 (c) -5, 3 ) (a) 0, -3 (b) 0, 4 (c), - 3) (a) -1/, 4/3 (b) 0.5,.5 4) (a) -5.30, (b) 1.07, (c) -1.0, 1.45 (d) no solutions (e) no solutions f) no solutions SECTION 8 Ex A y 1 1) x 7 Ex B 3rP 1) t w 3rP ) t w ) x 4y 5 3) x 3(4 y ) 3V 3) t h Pg 4) t 9y 0 4) x 1 Pag 5) t v w r a 6) t b Ex C c 3 1) x ) a b 3a k x k 3 3) y 3 x 5y 4) ab x b a SECTION 9 Ex A 1) 5b 6 ) 6c 7 3) b 3 c 4 4) -1n 8 5) 4n 5 6) d 7) a 6 8) -d 1 Ex B 1) ) 3 3) 1/3 4) 1/5 5) 1 6) 1/7 7) 9 8) 9/4 9) ¼ 10) 0. 11) 4/9 1) 64 13) 6a 3 14) x 15) xy

33 SECTION 10 Ex A 1) ) Ex B 1) ) 3) 3) 4) 4) 5) 6) 5) 6) 7) 8) Ex C 1) ) 3) 4) 5) 6) 7) 8) 9) 10) Ex D 1) ) 3) 4) 5) 6) 7) 8) 9) SECTION 11 Ex A 1) 8 ) -3 3) 56 4) 0 5) 65 6) Ex B 1) - ), 3

34 Solutions to the Practice Booklet Test 1 (a) (b) (c) (d) ) (a) 4x + 4x 3 (b) a + 6a + 9 (c) 10x -13x 3) (a) x(x 7) (b) (y + 8)(y 8) (c) (x - 1)(x + 3) (d) (3t - 5)(t 1) x 4) (a) y (b) 10 x 3 6 5) (a) h = 5 (b) x = 0 or x = 8 (c) p = -6 or p = 6) (a) x -4 (b) x 6 y 3 (c) x 7 7) (a) (b) 1 (c) 8) x = 3, y = 4 9) (a) v u x a (b) x 3V h (c) y x y 1 10) 1 1 x 10 11) (a) x < 1.6 (b) x < -.5 1) (a) 19 (b) 3 (c) (a) (b) (c) (d) (e)

MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS

MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS Introduction to A level Maths INDUCTION BOOKLET INTRODUCTION TO A LEVEL MATHS AT MGGS Thank you for choosing to study Mathematics in the sixth

More information

A Level Maths. Induction Booklet CONTENTS

A Level Maths. Induction Booklet CONTENTS A Level Maths Induction Booklet CONTENTS Chapter 1 Removing brackets page Chapter Linear equations page 4 Chapter 3 Simultaneous equations page 8 Chapter 4 Factors page 10 Chapter 5 Change the subject

More information

SIXTH FORM MATHEMATICS A LEVEL INDUCTION BOOKLET SEPTEMBER Name:

SIXTH FORM MATHEMATICS A LEVEL INDUCTION BOOKLET SEPTEMBER Name: SIXTH FORM MATHEMATICS A LEVEL INDUCTION BOOKLET SEPTEMBER 014 Name: INTRODUCTION TO A LEVEL MATHS Thank you for choosing to study Mathematics in the sixth form at Chelsea Academy. In year 1 you will sit

More information

MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS

MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS Introduction to A level Maths INDUCTION BOOKLET INTRODUCTION TO A LEVEL MATHS AT MGGS Thank you for choosing to study Mathematics in the sith

More information

Maths Department. A Level Induction Booklet

Maths Department. A Level Induction Booklet Maths Department A Level Induction Booklet One of the most important things if you are to succeed at A Level Maths is to ensure you understand all the algebra you met at GCSE. Working through the eamples

More information

Maths Department. A Level Induction Booklet

Maths Department. A Level Induction Booklet Maths Department A Level Induction Booklet CONTENTS Chapter 1 Removing brackets page Chapter Linear equations 4 Chapter 3 Simultaneous equations 8 Chapter 4 Factors 10 Chapter 5 Change the subject of the

More information

Alperton Community School. Preparation for. A Level Mathematics. This induction booklet is for students who wish to start AS Level Maths in Year 12.

Alperton Community School. Preparation for. A Level Mathematics. This induction booklet is for students who wish to start AS Level Maths in Year 12. Alperton Community School Preparation for A Level Mathematics This induction booklet is for students who wish to start AS Level Maths in Year 1. You are epected to know these topics before your first maths

More information

Introduction to A-Level Maths (Bridging Unit)

Introduction to A-Level Maths (Bridging Unit) Introduction to A-Level Maths (Bridging Unit) What is infinity + infinity? To infinity and beyond! SUMMER 017 Tuford Academy Faculty of Mathematics 1 INTRODUCTION TO A LEVEL MATHS AT TUXFORD ACADEMY Thank

More information

A-LEVEL MATHS Bridging Work 2017

A-LEVEL MATHS Bridging Work 2017 A-LEVEL MATHS Bridging Work 017 Name: Firstly, CONGRATULATIONS for choosing the best A-Level subject there is. A-Level Maths at Wales is not only interesting and enjoyable but is highly regarded by colleges,

More information

Preparing for A-Level Mathematics Summer 2017

Preparing for A-Level Mathematics Summer 2017 Preparing for A-Level Mathematics Summer 017 INTRODUCTION TO A LEVEL MATHS Thank you for choosing to study Mathematics in the sith form. You will sit two modules in Pure Mathematics (C1 and C) as well

More information

CONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION

CONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION CONTENTS CHECK LIST - - ACCURACY - 4 - FRACTIONS - 6 - INDICES - 9 - SURDS - - RATIONALISING THE DENOMINATOR - 4 - SUBSTITUTION - 5 - REMOVING BRACKETS - 7 - FACTORISING - 8 - COMMON FACTORS - 8 - DIFFERENCE

More information

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x

More information

Mathematics: Year 12 Transition Work

Mathematics: Year 12 Transition Work Mathematics: Year 12 Transition Work There are eight sections for you to study. Each section covers a different skill set. You will work online and on paper. 1. Manipulating directed numbers and substitution

More information

Expanding brackets and factorising

Expanding brackets and factorising Chapter 7 Expanding brackets and factorising This chapter will show you how to expand and simplify expressions with brackets solve equations and inequalities involving brackets factorise by removing a

More information

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name:

A Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name: A Level Summer Work Year 11 Year 12 Transition Due: First lesson back after summer! Name: This summer work is compulsory. Your maths teacher will ask to see your work (and method) in your first maths lesson,

More information

CHAPTER 1. Review of Algebra

CHAPTER 1. Review of Algebra CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you

More information

SUMMER REVIEW PACKET. Name:

SUMMER REVIEW PACKET. Name: Wylie East HIGH SCHOOL SUMMER REVIEW PACKET For students entering Regular PRECALCULUS Name: Welcome to Pre-Calculus. The following packet needs to be finished and ready to be turned the first week of the

More information

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is

More information

A-Level Maths Induction Summer Work

A-Level Maths Induction Summer Work A-Level Maths Induction Summer Work Name:. (Show workings for every question in this booklet) This booklet contains GCSE Algebra skills that you will need in order to successfully complete the A-Level

More information

Factoring and Algebraic Fractions

Factoring and Algebraic Fractions Worksheet. Algebraic Fractions Section Factoring and Algebraic Fractions As pointed out in worksheet., we can use factoring to simplify algebraic expressions, and in particular we can use it to simplify

More information

Factorisation CHAPTER Introduction

Factorisation CHAPTER Introduction FACTORISATION 217 Factorisation CHAPTER 14 14.1 Introduction 14.1.1 Factors of natural numbers You will remember what you learnt about factors in Class VI. Let us take a natural number, say 30, and write

More information

Intermediate Tier - Algebra revision

Intermediate Tier - Algebra revision Intermediate Tier - Algebra revision Contents : Collecting like terms Multiplying terms together Indices Expanding single brackets Expanding double brackets Substitution Solving equations Finding nth term

More information

P1 Chapter 1 :: Algebraic Expressions

P1 Chapter 1 :: Algebraic Expressions P1 Chapter 1 :: Algebraic Expressions jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 26 th August 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework

More information

A Level Maths summer preparation work

A Level Maths summer preparation work A Level Maths summer preparation work Welcome to A Level Maths! We hope you are looking forward to two years of challenging and rewarding learning. You must make sure that you are prepared to study A Level

More information

AS Maths for Maths Pack

AS Maths for Maths Pack Student Teacher AS Maths for Maths Pack September 0 City and Islington Sith Form College Mathematics Department www.candimaths.uk CONTENTS WS Numbers [Directed Numbers WS Numbers [Indices, powers WS Numbers

More information

Radicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).

Radicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize). Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:

More information

Table of Contents. Module 1

Table of Contents. Module 1 Table of Contents Module 1 1.1 Order of Operations 1.6 Signed Numbers 1. Factorization of Integers 1.7 Further Signed Numbers 1.3 Fractions 1.8 Power Laws 1.4 Fractions and Decimals 1.9 Introduction to

More information

Algebra Revision Guide

Algebra Revision Guide Algebra Revision Guide Stage 4 S J Cooper 1st Edition Collection of like terms... Solving simple equations... Factorisation... 6 Inequalities... 7 Graphs... 9 1. The straight line... 9. The quadratic curve...

More information

Algebra. Mathematics Help Sheet. The University of Sydney Business School

Algebra. Mathematics Help Sheet. The University of Sydney Business School Algebra Mathematics Help Sheet The University of Sydney Business School Introduction Terminology and Definitions Integer Constant Variable Co-efficient A whole number, as opposed to a fraction or a decimal,

More information

Section 2.4: Add and Subtract Rational Expressions

Section 2.4: Add and Subtract Rational Expressions CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall

More information

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self-paced Course MODULE ALGEBRA Module Topics Simplifying expressions and algebraic functions Rearranging formulae Indices 4 Rationalising a denominator

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

CAHSEE on Target UC Davis, School and University Partnerships

CAHSEE on Target UC Davis, School and University Partnerships UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 2006 Director Sarah R. Martinez,

More information

A-Level Notes CORE 1

A-Level Notes CORE 1 A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is

More information

WEST COVENTRY SIXTH FORM

WEST COVENTRY SIXTH FORM WEST COVENTRY SIXTH FORM West Coventry Academy SUBJECT TRANSITION BOOK Summer 2017 Mathematics STUDENT NAME: SCHOOL: This booklet has been prepared by maths staff for you to read and the work contained

More information

Chapter 8: Algebra Part 2

Chapter 8: Algebra Part 2 Chapter 8: Algebra Part 2 Section 8.1 Algebraic Products Expanding brackets means to remove the brackets. How would we expand the following? 5 (x + 2) The term which is outside the brackets must be multiplied

More information

Algebra Year 10. Language

Algebra Year 10. Language Algebra Year 10 Introduction In Algebra we do Maths with numbers, but some of those numbers are not known. They are represented with letters, and called unknowns, variables or, most formally, literals.

More information

Mathematics Revision Guide. Algebra. Grade C B

Mathematics Revision Guide. Algebra. Grade C B Mathematics Revision Guide Algebra Grade C B 1 y 5 x y 4 = y 9 Add powers a 3 a 4.. (1) y 10 y 7 = y 3 (y 5 ) 3 = y 15 Subtract powers Multiply powers x 4 x 9...(1) (q 3 ) 4...(1) Keep numbers without

More information

Sail into Summer with Math!

Sail into Summer with Math! Sail into Summer with Math! For Students Entering Algebra 1 This summer math booklet was developed to provide students in kindergarten through the eighth grade an opportunity to review grade level math

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Factorise each polynomial: a) x 2 6x + 5 b) x 2 16 c) 9x 2 25 2) Simplify the following algebraic fractions fully: a) x 2

More information

Algebra 2 Summer Work Packet Review and Study Guide

Algebra 2 Summer Work Packet Review and Study Guide Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the

More information

Maths A Level Summer Assignment & Transition Work

Maths A Level Summer Assignment & Transition Work Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first

More information

JUST THE MATHS UNIT NUMBER 1.5. ALGEBRA 5 (Manipulation of algebraic expressions) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.5. ALGEBRA 5 (Manipulation of algebraic expressions) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.5 ALGEBRA 5 (Manipulation of algebraic expressions) by A.J.Hobson 1.5.1 Simplification of expressions 1.5.2 Factorisation 1.5.3 Completing the square in a quadratic expression

More information

What you may need to do: 1. Formulate a quadratic expression or equation. Generate a quadratic expression from a description or diagram.

What you may need to do: 1. Formulate a quadratic expression or equation. Generate a quadratic expression from a description or diagram. Dealing with a quadratic What it is: A quadratic expression is an algebraic expression containing an x 2 term, as well as possibly an x term and/or a number, but nothing else - eg, no x 3 term. The general

More information

Finite Mathematics : A Business Approach

Finite Mathematics : A Business Approach Finite Mathematics : A Business Approach Dr. Brian Travers and Prof. James Lampes Second Edition Cover Art by Stephanie Oxenford Additional Editing by John Gambino Contents What You Should Already Know

More information

Geometry 21 Summer Work Packet Review and Study Guide

Geometry 21 Summer Work Packet Review and Study Guide Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

More information

Mathematics 1 Lecture Notes Chapter 1 Algebra Review

Mathematics 1 Lecture Notes Chapter 1 Algebra Review Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to

More information

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

P.1 Prerequisite skills Basic Algebra Skills

P.1 Prerequisite skills Basic Algebra Skills P.1 Prerequisite skills Basic Algebra Skills Topics: Evaluate an algebraic expression for given values of variables Combine like terms/simplify algebraic expressions Solve equations for a specified variable

More information

MATH98 Intermediate Algebra Practice Test Form A

MATH98 Intermediate Algebra Practice Test Form A MATH98 Intermediate Algebra Practice Test Form A MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y - 4) - (y + ) = 3y 1) A)

More information

Chapter 1 Review of Equations and Inequalities

Chapter 1 Review of Equations and Inequalities Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve

More information

Core 1 Basic Algebra. Section 1: Expressions and equations

Core 1 Basic Algebra. Section 1: Expressions and equations Core 1 Basic Algebra Section 1: Expressions and equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms Expanding brackets Factorising Multiplication

More information

3.1 Solving Quadratic Equations by Factoring

3.1 Solving Quadratic Equations by Factoring 3.1 Solving Quadratic Equations by Factoring A function of degree (meaning the highest exponent on the variable is ) is called a Quadratic Function. Quadratic functions are written as, for example, f(x)

More information

Dear Future Pre-Calculus Students,

Dear Future Pre-Calculus Students, Dear Future Pre-Calculus Students, Congratulations on your academic achievements thus far. You have proven your academic worth in Algebra II (CC), but the challenges are not over yet! Not to worry; this

More information

Unit 9 Study Sheet Rational Expressions and Types of Equations

Unit 9 Study Sheet Rational Expressions and Types of Equations Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by

More information

Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals

Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals Algebra 1 Math Review Packet Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals 2017 Math in the Middle 1. Clear parentheses using the distributive

More information

Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS:

Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS: Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS: 1 EXPONENT REVIEW PROBLEMS: 2 1. 2x + x x + x + 5 =? 2. (x 2 + x) (x + 2) =?. The expression 8x (7x 6 x 5 ) is equivalent to?.

More information

Solving Multi-Step Equations

Solving Multi-Step Equations 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract terms to both sides of the equation to get the

More information

Algebra One Dictionary

Algebra One Dictionary Algebra One Dictionary Page 1 of 17 A Absolute Value - the distance between the number and 0 on a number line Algebraic Expression - An expression that contains numbers, operations and at least one variable.

More information

2009 A-level Maths Tutor All Rights Reserved

2009 A-level Maths Tutor All Rights Reserved 2 This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents indices 3 laws of logarithms 7 surds 12 inequalities 18 quadratic

More information

6.1 Solving Quadratic Equations by Factoring

6.1 Solving Quadratic Equations by Factoring 6.1 Solving Quadratic Equations by Factoring A function of degree 2 (meaning the highest exponent on the variable is 2), is called a Quadratic Function. Quadratic functions are written as, for example,

More information

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y? REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate

More information

Summer Induction Work

Summer Induction Work A-level Maths Summer Induction Work Deadline: Monday 11th September The Hazeley Academy Mathematics A Level Maths Edecel: Pure Maths, Statistics and Mechanics Objective: To reinforce key GCSE skills in

More information

licensed to: St Andrews Academy Page 2

licensed to: St Andrews Academy Page 2 National 5 Mathematics Revision Notes Last updated January 014 Use this booklet to practise working independently like you will have to in the exam. Get in the habit of turning to this booklet to refresh

More information

September 12, Math Analysis Ch 1 Review Solutions. #1. 8x + 10 = 4x 30 4x 4x 4x + 10 = x = x = 10.

September 12, Math Analysis Ch 1 Review Solutions. #1. 8x + 10 = 4x 30 4x 4x 4x + 10 = x = x = 10. #1. 8x + 10 = 4x 30 4x 4x 4x + 10 = 30 10 10 4x = 40 4 4 x = 10 Sep 5 7:00 AM 1 #. 4 3(x + ) = 5x 7(4 x) 4 3x 6 = 5x 8 + 7x CLT 3x = 1x 8 +3x +3x = 15x 8 +8 +8 6 = 15x 15 15 x = 6 15 Sep 5 7:00 AM #3.

More information

ALGEBRA CLAST MATHEMATICS COMPETENCIES

ALGEBRA CLAST MATHEMATICS COMPETENCIES 2 ALGEBRA CLAST MATHEMATICS COMPETENCIES IC1a: IClb: IC2: IC3: IC4a: IC4b: IC: IC6: IC7: IC8: IC9: IIC1: IIC2: IIC3: IIC4: IIIC2: IVC1: IVC2: Add and subtract real numbers Multiply and divide real numbers

More information

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms

More information

Rising 8th Grade Math. Algebra 1 Summer Review Packet

Rising 8th Grade Math. Algebra 1 Summer Review Packet Rising 8th Grade Math Algebra 1 Summer Review Packet 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract

More information

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

More information

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2 Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

More information

Pure Mathematics P1

Pure Mathematics P1 1 Pure Mathematics P1 Rules of Indices x m * x n = x m+n eg. 2 3 * 2 2 = 2*2*2*2*2 = 2 5 x m / x n = x m-n eg. 2 3 / 2 2 = 2*2*2 = 2 1 = 2 2*2 (x m ) n =x mn eg. (2 3 ) 2 = (2*2*2)*(2*2*2) = 2 6 x 0 =

More information

Variables and Expressions

Variables and Expressions Variables and Expressions A variable is a letter that represents a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic

More information

Algebra I. Polynomials.

Algebra I. Polynomials. 1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying

More information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

Equations and Inequalities. College Algebra

Equations and Inequalities. College Algebra Equations and Inequalities College Algebra Radical Equations Radical Equations: are equations that contain variables in the radicand How to Solve a Radical Equation: 1. Isolate the radical expression on

More information

Algebra I. Book 2. Powered by...

Algebra I. Book 2. Powered by... Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........

More information

PLC Papers. Created For:

PLC Papers. Created For: PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use

More information

C if U can. Algebra. Name

C if U can. Algebra. Name C if U can Algebra Name.. How will this booklet help you to move from a D to a C grade? The topic of algebra is split into six units substitution, expressions, factorising, equations, trial and improvement

More information

Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.

Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc. Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a

More information

IES Parque Lineal - 2º ESO

IES Parque Lineal - 2º ESO UNIT5. ALGEBRA Contenido 1. Algebraic expressions.... 1 Worksheet: algebraic expressions.... 2 2. Monomials.... 3 Worksheet: monomials.... 5 3. Polynomials... 6 Worksheet: polynomials... 9 4. Factorising....

More information

GCSE Mathematics Non-Calculator Higher Tier Free Practice Set 1 1 hour 45 minutes ANSWERS. Grade Boundaries A* A B C D E.

GCSE Mathematics Non-Calculator Higher Tier Free Practice Set 1 1 hour 45 minutes ANSWERS. Grade Boundaries A* A B C D E. MathsMadeEasy GCSE Mathematics Non-Calculator Higher Tier Free Practice Set 1 1 hour 45 minutes ANSWERS Grade Boundaries A* A B C D E 88 71 57 43 22 13 3 Authors Note Every possible effort has been made

More information

Further Mathematics Summer work booklet

Further Mathematics Summer work booklet Further Mathematics Summer work booklet Further Mathematics tasks 1 Skills You Should Have Below is the list of the skills you should be confident with before starting the A-Level Further Maths course:

More information

June Dear Future Algebra 2 Trig Student,

June Dear Future Algebra 2 Trig Student, June 016 Dear Future Algebra Trig Student, Welcome to Algebra /Trig! Since we have so very many topics to cover during our 016-17 school year, it is important that each one of you is able to complete these

More information

Algebra 2 Segment 1 Lesson Summary Notes

Algebra 2 Segment 1 Lesson Summary Notes Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the

More information

Algebra I Polynomials

Algebra I Polynomials Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying

More information

Summer 2017 Math Packet

Summer 2017 Math Packet Summer 017 Math Packet for Rising Geometry Students This packet is designed to help you review your Algebra Skills and help you prepare for your Geometry class. Your Geometry teacher will expect you to

More information

Herndon High School Geometry Honors Summer Assignment

Herndon High School Geometry Honors Summer Assignment Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in

More information

Not drawn accurately

Not drawn accurately Q1. A trapezium has parallel sides of length (x + 1) cm and (x + 2) cm. The perpendicular distance between the parallel sides is x cm. The area of the trapezium is 10 cm 2. Not drawn accurately Find the

More information

Teddington School Sixth Form

Teddington School Sixth Form Teddington School Sixth Form AS / A level Maths Induction and Key Course Materials Introduction The Mathematics Department at Teddington School is delighted that you would like to continue your studies

More information

QUADRATIC EQUATIONS M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier

QUADRATIC EQUATIONS M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier Mathematics Revision Guides Quadratic Equations Page 1 of 8 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier QUADRATIC EQUATIONS Version: 3.1 Date: 6-10-014 Mathematics Revision Guides

More information

Core 1 Module Revision Sheet J MS. 1. Basic Algebra

Core 1 Module Revision Sheet J MS. 1. Basic Algebra Core 1 Module Revision Sheet The C1 exam is 1 hour 0 minutes long and is in two sections Section A (6 marks) 8 10 short questions worth no more than 5 marks each Section B (6 marks) questions worth 12

More information

Collecting Like Terms

Collecting Like Terms MPM1D Unit 2: Algebra Lesson 5 Learning goal: how to simplify algebraic expressions by collecting like terms. Date: Collecting Like Terms WARM-UP Example 1: Simplify each expression using exponent laws.

More information

Workshops: The heart of the MagiKats Programme

Workshops: The heart of the MagiKats Programme Workshops: The heart of the MagiKats Programme Every student is assigned to a Stage, based on their academic year and assessed study level. Stage 5 students have completed all Stage 4 materials. The sheets

More information

Roots of quadratic equations

Roots of quadratic equations CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients

More information

MA094 Part 2 - Beginning Algebra Summary

MA094 Part 2 - Beginning Algebra Summary MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

More information

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers

Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}

More information

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations

More information

Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.

Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200. Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

More information