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Equations and Inequations S4 Credit www.mathsrevision.com Reminder of Solving Equations 1 Reminder of Solving Equations 2 Equations with Fractions More Equations with Fractions Inequalities Solving Inequalities 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Starter Questions www.mathsrevision.com S4 Credit 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 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Reminder on Solving Equations www.mathsrevision.com S4 Credit Learning Intention 1. To remind pupils on how to solve various types of equations 29-Apr-17 Success Criteria 1. Understand the process of solving equations using “ the balancing method ” Created by Mr. Lafferty@mathsrevision.com Reminder on Solving Equations www.mathsrevision.com S4 Credit Multiply out the bracket first and then solve. Example 1 Substitute value into original equation to check answer 29-Apr-17 5(x - 3) = 25 5x - 15 = 25 5x =25 + 15 = 40 x = 40 ÷ 5 = 8 Created by Mr. Lafferty@mathsrevision.com Reminder on Solving Equations www.mathsrevision.com S4 Credit Example 2 Substitute value into original equation to check answer 18 18 = 3(x + 1) 6x - 12 = 3x + 3 6(x - 2) Solve as normal 6x – 3x -12 = 3 3x -12 = 3 3x = 3 + 12 3x = 15 x = 15 ÷ (3) = 5 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Equations and brackets S4 Credit www.mathsrevision.com Example 3 Substitute value into original equation to check answer 6 6 -3(2 + 2y) = 6 – (y + 2) -6 - 6y = 6 - y - 2 -6 – 6y = 4 - y -6 - 6y + y = 4 -6 – 5y = 4 Tidy up RHS Solve as normal – 5y = 10 y = 10 ÷ (-5) = -2 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Reminder on Solving Equations S4 Credit www.mathsrevision.com Now try Ex 2.1 Ch5 MIA (page 99) 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Starter Questions www.mathsrevision.com S4 Credit Q1. Explain why the following are True or False (a) -3y Q2. Is the following true x 5y = -30y (b) -6q x (-4q) = 24q2 -2a( b – a) = -2ab +4a Q3. Write down the two numbers that multiply to give 8 and subtract to give 2. 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Reminder on Solving Equations www.mathsrevision.com S4 Credit Learning Intention 1. To remind pupils on how to solve equations containing square terms. 29-Apr-17 Success Criteria 1. Understand the process of solving equations with square terms using “ the balancing method ” Created by Mr. Lafferty@mathsrevision.com Equations and brackets www.mathsrevision.com S4 Credit Multiply out the bracket first 289 and then solve. 289 Example 1 (x + 2)2 = x2 + 82 FOIL Substitute value (x + 2)(x + 2) = x2 + 64 into original equation to x2 + 4x + 4 = x2 + 64 check answer 4x + 4 = 64 4x = 60 x = 60 ÷ 4 = 15 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Equations and brackets www.mathsrevision.com S4 Credit Multiply out the bracket first and then solve. 35 Example 2 Substitute value into original equation to check answer 29-Apr-17 m(m + 2) m2 + 2m 2m m 35 = m2 + 10 = m2 + 10 = 10 = 10 ÷ 2 = 5 Created by Mr. Lafferty@mathsrevision.com Equations and brackets S4 Credit www.mathsrevision.com Example 5 : The two areas are equal. Find the value of x. (x + 2) A = (x + A = (x + 7)(x – 2) (x – 2) 2)2 400 (x + 2) Substitute value into original equation to check answer 29-Apr-17 (x + 7) 400 (x + 2)2 = (x + 7) (x - 2) Foil x2 + 4x + 4 = x2 + 5x - 14 Solve in the 4x + 4 = 5x - 14 usual way 4 = x - 14 x = 18 Created by Mr. Lafferty@mathsrevision.com Equations & Pythagoras S4 Credit Example www.mathsrevision.com Using Pythagoras find the length of all sides of the triangle. Multiplication Table 25 25 (x + 3)2 = (x + 2)2 + 32 x2 + 6x + 9 = x2 + 4x + 13 Balancing 6x + 9 = 4x + 13 Method 2x = 4 x=2 Check ! 5 x + 3 cm 3cm x + 2 cm 4 Lengths are 3 , 4 and 5 cm Equations and brackets S4 Credit www.mathsrevision.com Now try Ex 2.2 Ch5 MIA (page 101) 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Starter Questions S4 Credit www.mathsrevision.com 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? 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Equations & Fractions Fractional Equations with Brackets www.mathsrevision.com S4 Credit Learning Intention Success Criteria 1. To show how to solve fractional equations using all the rules learned so far. 29-Apr-17 1. Apply Balancing Method to solve fractional equations. Created by Mr. Lafferty Maths Dept. Equations & Fractions S4 Credit www.mathsrevision.com Process of solving equations with fractions 1. 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. 29-Apr-17 Created by Mr. Lafferty Maths Dept. Equations & Fractions Equations with Fractions www.mathsrevision.com S4 Credit Multiply EVERY term by 5 29-Apr-17 y 1 5 3 y 1 5 5 5 3 5 y 3 Created by Mr. Lafferty Maths Dept. Equations & Fractions Equations with Fractions www.mathsrevision.com S4 Credit Multiply EVERY term by 3 Balancing Method 7 m (m 2) 3 3m 7(m 2) 3m 7 m 14 4m 14 7 m 2 Remove brackets Equations & Fractions Equations with Fractions www.mathsrevision.com S4 Credit Multiply EVERY term by 4 Balancing Method 3 x2 x 4 3 4x 8 4 x 4 4 x 8 3x x 8 Tidy up Equations & Fractions S4 Credit www.mathsrevision.com Now try Ex 3.1 Ch5 MIA (page 103) 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Starter Questions www.mathsrevision.com S4 Credit Q1. Find the area of the second shape given the first has area 7w - 14 (w - 2) (w – 5) 7 3 Q2. Split 48 into the ratio 1:3 Q3. In standard form 18294000 is 1.8x107 is this correct? 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Equations & Fractions www.mathsrevision.com S4 Credit Learning Intention 1. To remind pupils on how to solve equations containing fraction terms. 29-Apr-17 Success Criteria 1. Understand the process of solving equations with fraction terms using “ the balancing method ” Created by Mr. Lafferty@mathsrevision.com Equations & Fractions www.mathsrevision.com S4 Credit Multiply EVERY term by LCM 12 Equations with Fractions x x 3 4 6 x x 12 12 12 3 4 6 3x 2 x 36 Balancing Method x 36 29-Apr-17 Created by Mr. Lafferty Maths Dept. Tidy up Equations & Fractions www.mathsrevision.com S4 Credit Multiply EVERY term by LCM 6 Equations with Fractions y 1 y 1 2 2 3 Tidy up ( y 1) ( y 1) 6 6 6 2 2 3 Balancing 3( y 1) 2( y 1) 12 29-Apr-17 Created by Mr. Lafferty Maths Dept. Method Equations & Fractions Equations with Fractions www.mathsrevision.com S4 Credit 3( y 1) 2( y 1) 12 3 y 3 2 y 2 12 y 5 12 y 17 29-Apr-17 Created by Mr. Lafferty Maths Dept. Balancing Method Equations & Fractions S4 Credit www.mathsrevision.com Now try Ex 4.1 Ch5 MIA (page 106) 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Starter Questions S4 Credit www.mathsrevision.com Q1. Solve for x (a) x+3=8 (b) 2x – 14 = 50 Q2. Is this statement true (x – 1) – 3(x + 1) = -2x Q3. 29-Apr-17 1 1 3 1 2 3 Created by Mr. Lafferty@mathsrevision.com Equations & Inequalities www.mathsrevision.com Solving Inequalities Learning Intention 1. To show how we can solve inequalities using the same rules we use for equations. 29-Apr-17 Success Criteria 1. Understand the term inequality. 2. Solve inequalities using the same method as equations. Created by Mr. Lafferty Maths Dept. Equations & Inequalities Solving Inequalities www.mathsrevision.com The Good News Inequalities are similar to equations except we replace the “=“ with one of the following symbols : " < " " > " " " or Less than Greater than 29-Apr-17 Less than or equal to Created by Mr. Lafferty Maths Dept. "" Greater than or equal to Removing a Single Bracket S4 Credit www.mathsrevision.com Now try Ex 5.1 Ch5 MIA (page 108) 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com Starter Questions S4 Credit www.mathsrevision.com Q1. Does 579300 = 5.793x105 Q2. Explain why the answer to 4(w + 2) = 6(w + 1) is w = 1 Q3. 29-Apr-17 1 3 7 2 4 Created by Mr. Lafferty@mathsrevision.com Equations & Inequalities www.mathsrevision.com Solving Inequalities Learning Intention 1. To show how we can solve inequalities using the same rules we use for equations. 29-Apr-17 Success Criteria 1. Understand the term inequality. 2. Solve inequalities using the same method as equations. Created by Mr. Lafferty Maths Dept. Equations & Inequalities Solving Inequalities www.mathsrevision.com The Good News Inequalities are similar to equations except we replace the “=“ with one of the following symbols : " < " " > " " " or Less than Greater than 29-Apr-17 Less than or equal to Created by Mr. Lafferty Maths Dept. "" Greater than or equal to Equations & Inequalities Balancing Method www.mathsrevision.com Add 1 to both sides Solving Inequalities Even Better News ! Solving inequalities is almost identical to solving equations : 2x 1 7 2x 7 1 2x 8 x4 29-Apr-17 Tidy up Divide by 2 both sides x is any value less than 4 Created by Mr. Lafferty Maths Dept. Equations & Inequalities Multiply out brackets Solving Inequalities www.mathsrevision.com Solving inequalities is almost identical to solving equations : Add 6 to both side Divide both sides by 3 2(2 x 3) x 9 4x 6 x 9 Subtract x from each side. 4 x x 15 x 15 3 3 x 15 x 5 x is any value greater than or equal to 5 29-Apr-17 Created by Mr. Lafferty Maths Dept. Inequalities Solving Inequalities www.mathsrevision.com S4 Credit 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 Divide across by -3 and change the Sign -3 So m > 2 Inequalities Solving Inequalities S4 Credit www.mathsrevision.com Example 2 5( x – 1 ) - 8x ≥ - 17 5x – 5 – 8x ≥ - 17 - 3x - 5 ≥ - 17 - 3x ≥ - 12 x 29-Apr-17 ≤ -12 -3 So x ≤ 4 Inequalities www.mathsrevision.com S4 Credit Now try 6.1 & 6.2 Ch5 MIA (page 110) 29-Apr-17 Created by Mr. Lafferty@mathsrevision.com