Junior Certificate Ordinary Level Algebra Practice Exercises

Junior Certificate Ordinary Level Algebra www.mathsplus.ie www.mathsplus.ie www.mathsplus.ie

(a) If x = 3, find the value of x2 2x + 5. 2x + 5 2 x (3)2 2(3) + 5 = 9 6 + 5 = 8

(b) (i) Multiply (2x + 1) by (3x 5) and write your answer in its simplest form. (2x + 1)(3x 5) 6x2 10x + 3x 7x 5

(b) (ii) Graph on the number line the solution set of 5x 1 14, x . 14 + 1, 2, 3, 4.. 5x 1 5x 15 Natural number Divide both sides by 5 x 3 -1 1 2 3 0 4

(a) Solve the equation 3(x 1) = 12. Expand the brackets 3(x 1) = 12 3x 3x = 15 Divide both sides by 3 x = 5 = 12+ 3 Collect like terms

(b) (i) Factorise 2xy 4xw 2xy 2x 4xw (y 2w) (ii) ab 2ac + 3b 6c + 3b 6c 2ac ab a(b 2c) + 3(b 2c) (a + 3)(b 2c)

(b) (iii) Factorise x2+ 2x 8 x2+ 2x 8 Factors must subtract to give 2 Factors of 8 Factors of 1 ( ) (x + 4) (x (x 2) ? ? + 4 2) 1 x 1 2 ( ) 1 x 8 4 (x 4) 1 = 7 2 = 2 8

(b) (iv) Factorise 36 y2 36 y2 62 a2 b2= (a b)(a + b) (6 y)(6 + y)

(c) x is a number. A second number is 5 greater than x. (i) Write down the second number in terms of x. x + 5 x+ 3( = 35 2 ) (ii) Twice the first number added to three times the second number is equal to 35. Write down an equation in x to represent this information. (iii) Solve your equation for x and state what the two numbers are. 2x + 3x 15 5x + x 4 = = 20 35 4 First number = Second number = 9

(c) x is a number. A second number is 5 greater than x. (iv) Verify your result. x+3( = 35 x + 5 2 ) 2(4) + 3(4 + 5) = 35 8 + 3(9) = 35 8 + 27 = 35 First number = Second number = 4 9

(a) If a = 2 and b = 7, find the value of: (i) 2a + b (2) 2a + = 4 + 7 = 11 b 7 (ii) 3ab + 1 + = 42 + 1 = 43 (7) ab 3 (2) 1

(b) (i) Solve the equation 5(2x + 1) = 45. 5(2x + 1) = 45 10x + = 5 x = 4 40 45

(b) (ii) Write in its simplest form Expand the brackets (6x y) 3(x 2y + 1). Collect like terms 3(x 2y + 1) 3x 3 + 5y ( ) y 6x 6x y + 3 6y 3x

(a) Find the values of x for which 2x + 1 7, x . 2x + 1 7 Natural number 1, 2, 3, 4.. 2 6 x x 3 Divide both sides by 2 x = {1, 2, 3}

(b) (i) Factorise 3x 3y + ax ay. + ax ay 3x 3y 3(x y)+ a(x y) (3 + a)(x y)

(b) (ii) Factorise x2 25. 52 x2 25 a2 b2= (a b)(a + b) (x 5)(x + 5)

(c) (i) Solve the equation x2 3x 10 = 0. x2 3x 10 = 0 Factors must subtract to give 3 Factors of 10 Factors of 1 ( ) (x + 2) + (x ? ? 5 = 0 2) (x 5) 1 1 x 1 2 5 x 10 x + 2 = 0 x = 2 x 5 = 0 5) ( ) (x x = 5 1 = 9 2 = 3 10

(c) (ii) Multiply (x 4) by (x2+ 3x 1) and write your answer in its simplest form. (x 4)(x2+ 3x 1) x 3+ + 4 13x 3x2 x2 4x2 12x x

(a) If x = 4, find the value of: (i) 5x + 3 + 3 = 20 + 3= 23 5x (4) (ii) x2 x + 7 (4)2 x + 7 (4) = 16 4 + 7 = 19 x2

(b) (i) Multiply (3x 2) by (4x + 5) and write your answer in its simplest form. (3x 2)(4x + 5) 12x2+ 15x + 7x 8x 10

(b) (ii) Write in its simplest form (4x2 3x + 7) + (x2 2x 8). ( ) ( ) + + 4x2 3x 7 2x 8 x2 5x2 5x 1

(c) A rectangle has a length (x + 6) cm and width x cm, as in the diagram. x + 6 (i) Find the perimeter of this rectangle in terms of x. x (ii) If the perimeter of the rectangle is 40 cm, write down an equation in x to represent this information. x + (x + 6) + x 12 + (x + 6) = 40 = 28 x = 7 cm (iii) Solve the equation that you formed in part (ii) above, for x. + 4x

(c) A rectangle has a length (x + 6) cm and width x cm, as in the diagram. x + 6 (i) Find the perimeter of this rectangle in terms of x. Perimeter 4l = 40 cm l = 10 cm Area = l2 x l (ii) If the perimeter of the rectangle is 40 cm, write down an equation in x to represent this information. = 102 = 100 cm2 (iii) Solve the equation that you formed in part (ii) above, for x. (iv) Find the area of the square with the same perimeter as the given rectangle. Give your answer in cm2.

(a) Solve the equation 5x 6 = 3(x + 4). 5x 6 = 3(x + 4) 5x = 6 2x = 18 x = 9 + 3x 12

(b) (i) Factorise 4ab + 8b + 8b 4ab (a + 2) 4 b

(b) (iii) Factorise x2+ 2x 15 x2+ 2x 15 Factors must subtract to give 2 Factors of 15 Factors of 1 ( ) (x + 5) (x 3) (x ? ? + 5 3) 1 x 3 1 ( ) 1 x 15 5 (x 5) 1 = 14 3 = 2 15

(b) (iv) Factorise x2 y2 Looney-Tunes-cn15.jpg x2 y2 a2 b2= (a b)(a + b) (x y)(x + y)

(b) (ii) Multiply (x 5) by (2x + 3). Write your answer in its simplest form. (x 5)(2x + 3) 7x 10x 2x2+ 3x 15

(a) Write in its simplest form 4(x + 3) + 2(5x + 4). 4(x + 3) + 2(5x + 4) 4x + + 12 14x + 20 + 10x 8

(b) Factorise (i) xy + wy + wy xy y(x + w)

(b) Factorise (ii) ax ay + bx by + bx by ax ay a(x y) b(x y) (a + b)(x y)

(b) (iii) Factorise p2 36. 62 p2 36 a2 b2= (a b)(a + b) (p 6)(p + 6)

(b) Factorise (iv) 4a2+ 8a + 4a2 8a 4 a a a 4 2 a ( ) a 2 + 2 4a

(c) (i) Solve the equation x2 5x 14 = 0. x2 5x 14 = 0 Factors must subtract to give 5 Factors of 14 Factors of 1 ( ) (x (x + 2) ? 7 + = 0 2) (x 7) x 7 = 0 x = 7 1 x 1 2 ? x + 2 = 0 x = 2 ( ) 1 x 14 7 (x 7) 1 = 13 2 = 5 14

(a) If x = 3, find the value of: (i) 4x + 5 + = 12 + 5 = 17 Powers first 4x (3) 5 (ii) 2x2 11 2 = 2(9) 11 = 18 11 = 7 2x 11 (3)

(b) (i) Solve the equation 4(5x + 6) = 84. 4(5x + 6) = 84 20x + 24 = 84 60 x = 3

(b) (ii) Write in its simplest form Expand the brackets 3x2 2x + 6 x(2x 3). Collect like terms 2x + 6 x(2x 3) 2x2 + 6 3x2 3x2 2x + 6 + 3x + x x2

(c) (i) Liam drove from Town A to Town B, a distance of x km. He then drove from Town B to Town C, a distance of (2x + 1) km. The total distance that he drove was 56 km. Find the value of x, correct to the nearest kilometre. x + (2x + 1) 3x = 56 + 3 1= 56 55 = 18 333 Divide both sides by 3 km

(a) Find the values of x for which 3x + 2 < 11, x . 3x + 2 <11 Natural number 1, 2, 3, 4.. 3 < 9 x x < 3 x = {1, 2} Divide both sides by 3

(b) (i) Factorise 16xy + 11y. + 11y 16xy y(16x + 11)

(b) (ii) Factorise 5x + 10y + ax + 2ay. + ax + 2ay + 10y 5x 5(x + 2y) + a(x + 2y) (5 + a)(x + 2y)

1 (b) (iii) Factorise x2 x 90. x2 x 90 Factors must subtract to give 1 Factors of 90 Factors of 1 ( ) (x + 9) (x ? ? + 9) (x 10) 1 x 5 9 ( ) 1 x 18 10 (x 10) 5 = 13 9 = 1 10 18

(b) (iv) Factorise x2 121. 112 x2 121 a2 b2= (a b)(a + b) (x 11)(x + 11)

(c) (iii) Solve the equation x2+ 5x 36 = 0. x2+ 5x 36 = 0 Factors must subtract to give 5 Factors of 36 Factors of 1 ( ) (x +9) x + 9 = 0 x = 9 = 0 (x (x 4) ? ? + 9 4) 1 x 2 4 x 4 = 0 x = 4 ( ) 1 x 18 9 (x 9) 2 = 16 4 = 5 18

(a) If a = 5 and b = 7, find the value of: (i) 9a + b a (5) + 9 = 45 + 7= 52 b7 (ii) ab + 13 + = 35 + 13 = 48 (5) ab (7) 13

(b) (i) Solve the equation 3(2x 1) = 4x + 9. 3(2x 1) = 4x + 9 6x = 3 2x = 12 x = 6 + 4x 9

(b) (ii) Multiply (5x 2) by (3x + 4) and write your answer in its simplest form. (5x 2)(3x + 4) 15x2+ + 14x 6x 20x 8

(c) (i) Shane is x years old. Eileen is three years younger than Shane. Find Eileen's age in terms of x. x 3 (ii) If the sum of Shane s age and Eileen s age is 47, write down an equation in x to represent this information. x + x 3= 47 Solve the equation that you formed in part (ii) above, for x. 2x =47 + 3 x = 25 2 (iii) 50

(c) (iv) Shane is x years old. Eileen is three years younger than Shane. When Eileen is 2x + 5 years old, find the sum of Shane s age and Eileen s age. 2x + 5 =55 Eileen +___ 2(25) + 5 58 Shane 113 x = 25

(a) Find the values for which 3x + 2 8, x . 3x + 2 Divide both sides by 3 x 2 x = {1, 2} 8 6 Natural number 1, 2, 3, 4..