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Equations and Inequations

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  1. Equations and Inequations Reminder of Solving Equations 1 Reminder of Solving Equations 2 Equations with Fractions More Equations with Fractions Inequalities Solving Inequalities Created by Mr. Lafferty@mathsrevision.com

  2. Starter Questions Q1. Is the following true or false (A) (-3a) x 5a = -15a (B) (-6x) x (-7y) =-42xy Q2. Are the 2 answers the same ? (a) (-3h)2 = (b) -(3d)2 = Q3. Spilt £64 pounds into the ratio 7:9 Q4. Explain why 0.0675 is equal to 6.75x10-2 Created by Mr. Lafferty@mathsrevision.com

  3. Reminder on Solving Equations Learning Intention Success Criteria • To remind pupils on how to solve various types of equations • Understand the process of solving equations using • “ the balancing method ” Created by Mr. Lafferty@mathsrevision.com

  4. Reminder on Solving Equations Multiply out the bracket first and then solve. Example 1 25 5(x - 3) = Substitute value into original equation to check answer 5x - 15 = 25 = 40 5x = 25 + 15 x = 40 ÷ 5 = 8 Created by Mr. Lafferty@mathsrevision.com

  5. Reminder on Solving Equations 18 18 Example 2 3(x + 1) 6(x - 2) = Solve as normal Substitute value into original equation to check answer = 3x + 3 - 12 6x = 3 6x – 3x -12 = 3 3x -12 = 3 + 12 3x = 15 3x = 5 = 15 ÷ (3) x Created by Mr. Lafferty@mathsrevision.com

  6. Equations and brackets 6 6 Example 3 Tidy up RHS 6 – (y + 2) -3(2 + 2y) = Substitute value into original equation to check answer - y - 2 = 6 - 6y -6 Solve as normal = 4 - y -6 – 6y = 4 -6 - 6y + y = 4 -6 – 5y = 10 – 5y = -2 = 10 ÷ (-5) y Created by Mr. Lafferty@mathsrevision.com

  7. Reminder on Solving Equations Now try Ex 2.1 Ch5 MIA (page 99) Created by Mr. Lafferty@mathsrevision.com

  8. Starter Questions Q1. Explain why the following are True or False (a) -3y x 5y = -30y (b) -6q x (-4q) = 24q2 Q2. Is the following true -2a( b – a) = -2ab +4a Q3. Write down the two numbers that multiply to give 8 and subtract to give 2. Created by Mr. Lafferty@mathsrevision.com

  9. Reminder on Solving Equations Learning Intention Success Criteria • To remind pupils on how to solve equations containing square terms. • Understand the process of solving equations with square terms using • “ the balancing method ” Created by Mr. Lafferty@mathsrevision.com

  10. Equations and brackets Multiply out the bracket first and then solve. 289 289 Example 1 x2 + 82 FOIL (x + 2)2 = Substitute value into original equation to check answer (x + 2)(x + 2) = x2 + 64 x2 + 4x + 4 = x2 + 64 = 64 4x + 4 4x = 60 x = 60 ÷ 4 = 15 Created by Mr. Lafferty@mathsrevision.com

  11. Equations and brackets Multiply out the bracket first and then solve. 35 35 Example 2 m2 + 10 m(m + 2) = Substitute value into original equation to check answer m2 + 2m = m2 + 10 2m = 10 m = 10 ÷ 2 = 5 Created by Mr. Lafferty@mathsrevision.com

  12. Equations and brackets Example 5 : The two areas are equal. Find the value of x. A = (x + 7)(x – 2) (x – 2) (x + 2) A = (x + 2)2 (x + 7) 400 400 Foil (x + 2) (x + 2)2 = (x + 7) (x - 2) Substitute value into original equation to check answer = x2 + 5x - 14 x2 + 4x + 4 Solve in the usual way 4x + 4 = 5x - 14 4 = x - 14 x = 18 Created by Mr. Lafferty@mathsrevision.com

  13. Equations & Pythagoras x + 3 cm 3cm x + 2 cm Example Using Pythagoras find the length of all sides of the triangle. 25 25 5 Multiplication Table (x + 3)2 = (x + 2)2 + 32 = x2 + 4x + 13 x2 + 6x + 9 Balancing Method 6x + 9 = 4x + 13 4 2x = 4 x = 2 Lengths are 3 , 4 and 5 cm Check !

  14. Equations and brackets Now try Ex 2.2 Ch5 MIA (page 101) Created by Mr. Lafferty@mathsrevision.com

  15. Starter Questions Q1. Multiply out. (a) 3y(x - y) = (b) 6q2 (2 - 4q) = Q2. Explain your working to show that if we split 24 into the ratio 1:5 the answer is 4:20 Q3. Writing out in full 5.2 x 10-3 to get 0.052 Is the correct? Created by Mr. Lafferty@mathsrevision.com

  16. Equations & Fractions Fractional Equations with Brackets Learning Intention Success Criteria 1. Apply Balancing Method to solve fractional equations. • 1. To show how to solve fractional equations using all the rules learned so far. Created by Mr. Lafferty Maths Dept.

  17. Equations & Fractions Process of solving equations with fractions • Remove any fraction by multiplying each term by the denominator value. • 2. Simplify each side of the equation if possible. • 3. Carry out balancing method to solve equation. Created by Mr. Lafferty Maths Dept.

  18. Equations & Fractions Equations with Fractions Multiply EVERY term by 5 Created by Mr. Lafferty Maths Dept.

  19. Equations & Fractions Equations with Fractions Multiply EVERY term by 3 Remove brackets Balancing Method

  20. Equations & Fractions Equations with Fractions Multiply EVERY term by 4 Tidy up Balancing Method

  21. Equations & Fractions Now try Ex 3.1 Ch5 MIA (page 103) Created by Mr. Lafferty@mathsrevision.com

  22. Starter Questions Q1. Find the area of the second shape given the first has area 7w - 14 (w - 2) (w – 5) 3 7 Q2. Split 48 into the ratio 1:3 Q3. In standard form 18294000 is 1.8x107 is this correct? Created by Mr. Lafferty@mathsrevision.com

  23. Equations & Fractions Learning Intention Success Criteria • To remind pupils on how to solve equations containing fraction terms. • Understand the process of solving equations with fraction terms using • “ the balancing method ” Created by Mr. Lafferty@mathsrevision.com

  24. Equations & Fractions Equations with Fractions Multiply EVERY term by LCM 12 Tidy up Balancing Method Created by Mr. Lafferty Maths Dept.

  25. Equations & Fractions Equations with Fractions Multiply EVERY term by LCM 6 Tidy up Balancing Method Created by Mr. Lafferty Maths Dept.

  26. Equations & Fractions Equations with Fractions Balancing Method Created by Mr. Lafferty Maths Dept.

  27. Equations & Fractions Now try Ex 4.1 Ch5 MIA (page 106) Created by Mr. Lafferty@mathsrevision.com

  28. Starter Questions Q1. Solve for x (a) x + 3 = 8 (b) 2x – 14 = 50 Q2. Is this statement true (x – 1) – 3(x + 1) = -2x Q3. Created by Mr. Lafferty@mathsrevision.com

  29. Solving Inequalities Equations & Inequalities Learning Intention Success Criteria 1. Understand the term inequality. • 1. To show how we can solve inequalities using the same rules we use for equations. 2. Solve inequalities using the same method as equations. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  30. Equations & Inequalities Solving Inequalities The Good News Inequalities are similar to equations except we replace the “=“ with one of the following symbols : www.mathsrevision.com Less than Greater than or equal to Less than or equal to Greater than Created by Mr. Lafferty Maths Dept.

  31. Removing a Single Bracket Now try Ex 5.1 Ch5 MIA (page 108) Created by Mr. Lafferty@mathsrevision.com

  32. Starter Questions Q1. Does 579300 = 5.793x105 Q2. Explain why the answer to 4(w + 2) = 6(w + 1) is w = 1 Q3. Created by Mr. Lafferty@mathsrevision.com

  33. Solving Inequalities Equations & Inequalities Learning Intention Success Criteria 1. Understand the term inequality. • 1. To show how we can solve inequalities using the same rules we use for equations. 2. Solve inequalities using the same method as equations. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  34. Equations & Inequalities Solving Inequalities The Good News Inequalities are similar to equations except we replace the “=“ with one of the following symbols : www.mathsrevision.com Less than Greater than or equal to Less than or equal to Greater than Created by Mr. Lafferty Maths Dept.

  35. Equations & Inequalities Balancing Method Add 1 to both sides Solving Inequalities Even Better News ! Solving inequalities is almost identical to solving equations : Tidy up www.mathsrevision.com Divide by 2 both sides x is any value less than 4 Created by Mr. Lafferty Maths Dept.

  36. Equations & Inequalities Multiply out brackets Solving Inequalities Solving inequalities is almost identical to solving equations : Add 6 to both side Subtract x from each side. Divide both sides by 3 www.mathsrevision.com x is any value greater than or equal to 5 Created by Mr. Lafferty Maths Dept.

  37. Inequalities Solving Inequalities The only one to watch out for is when you are dividing by a negative Example 1 8 – 3m < 2 -3m < -6 Subtract 8 from each side m -6 -3 > Divide across by -3 and change the Sign So m > 2

  38. Inequalities Solving Inequalities Example 2 5( x – 1 ) - 8x ≥ - 17 5x – 5 – 8x ≥ - 17 - 3x - 5 ≥ - 17 - 3x ≥ - 12 x -12 -3 ≤ So x ≤ 4

  39. Inequalities Now try 6.1 & 6.2 Ch5 MIA (page 110) Created by Mr. Lafferty@mathsrevision.com