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1 PROPERTIES OF EQUALITY Standard 1 SOLUTION OF EQUATIONS INCLUDING ABSOLUTE VALUE EQUATIONS EQUATIONS LEVEL 1 EQUATIONS LEVEL 2 EQUATIONS LEVEL 3 EQUATIONS.

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Presentation on theme: "1 PROPERTIES OF EQUALITY Standard 1 SOLUTION OF EQUATIONS INCLUDING ABSOLUTE VALUE EQUATIONS EQUATIONS LEVEL 1 EQUATIONS LEVEL 2 EQUATIONS LEVEL 3 EQUATIONS."— Presentation transcript:

1 1 PROPERTIES OF EQUALITY Standard 1 SOLUTION OF EQUATIONS INCLUDING ABSOLUTE VALUE EQUATIONS EQUATIONS LEVEL 1 EQUATIONS LEVEL 2 EQUATIONS LEVEL 3 EQUATIONS LEVEL 4 EQUATIONS LEVEL 5 ABSOLUTE VALUE DEFINITION ABSOLUTE VALUE EQUATIONS END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 2 Standard 1: Students solve equations and inequalities involving absolute value. Estándar 6: Los estudiantes resuelven ecuaciones y desigualdades que involucran valor absoluto. ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 3 PROPERTIES OF EQUALITY Standard 1 REFLEXIVE PROPERTY OF EQUALITY: For any real number a, a=a 5=5 -10=-10 SYMMETRIC PROPERTY OF EQUALITY: For all real numbers a and b, if a=b, then b=a X=5 5=X 6X-12=8 8=6X-12 9Y -2Y +1= 3X 2 3X= 9Y -2Y+1 2 SUBSTITUTION PROPERTY OF EQUALITY: If a=b, then a may be replaced by b. b=2 and3b +1=7 If then 3( )+1=7 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 4 TRANSITIVE PROPERTY OF EQUALITY: For all real numbers a, b, and c, if a=b, and b=c then a=c If X=6 and Y= 6 then X=Y If Y=2X+2 and Y=6-3X then 2X+2=6-3X ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY: PROPERTIES OF EQUALITY Standard 1 For any numbers a, b, and c, if a=b then a+c=b+c and a-c=b-c 10 = 10 + 6 +6 16 = 16 22 = 22 -5 17 = 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 5 MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY: PROPERTIES OF EQUALITY Standard 1 For any real numbers a, b, and c, if a=b, then a c=b c and if c=0, = a c b c 15 = 15 2 2 30 = 30 28 = 28 7 4 = 4 24 = 24 3 3 72 = 72 36 = 36 12 3 = 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 6 Standard 1 Solve these equations: X + 2 = 10 X - 6 = 15 8 + Y = 11 23 = 10 + K 12 = 9 + R 7 = Z -34-J +8 = 26 -12 = -9 -W 5X = 10 12 = 4Y 2X = -20-3F =18 -2 X = 8 +6 X = 21 -8 Y = 3 -10 13 = K K =13 -9 -9 3 = R R = 3 +34 41 = Z Z = 41 -8 -J = 18 (-1) J = -18 +9 -3 = -W (-1) 3 = W W = 3 5 X = 2 4 3 = Y Y = 3 2 X = -10 -3 F = -6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 7 Standard 1 K 9 = 5 R 2 = -8 X 4 6= Y -5 3= Z -3 -2 = +2 66=6X 6 6 X=11 5 + 2Z = 13-12 = 4 -8X 64 = 6X- 2 10X + 20 = 90 -20 10X = 70 10 X=7 (9) K = 45 (2) R = -16 (4) 24 =X X = 24 (-5) -15 = Y Y = -15 6 = Z (-3) Z = 6 -5 2Z = 8 2 2 Z = 4 -4 -16 = -8X -8 X = 2 2 = X K 9 = 5 R 2 = -8 X 4 6= Y -5 3= Z -3 -2 = Solve these equations: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 8 Standard 1 – 4 = 12 R 3 2 + = 13 Y 5 + 1 = 17 X 2 5 - = 23 Y 4 6 -2X 4 8 = 5X - 3 2 = 6 X= 3 3 7 X 2 =16 (2) X = 32 +4 R 3 =16 (3) R = 48 -2 Y 5 =11 (5) Y = 55 -5 -Y 4 =18 Y = -72 (-4) (2) 5X – 3= 12 +3 5X = 15 5 X = 3 (4) 32 = 6 – 2X -6 26 = -2X -2 -13 = X X = -13 R= 2 2 3 3 2 3 2 R = 3 +1 X – 1 = 2 3 7 5X - 3 2 = 6 6 -2X 4 8 = R= 2 2 3 7 3 7 3 X = 7 Solve these equations: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 9 Standard 1 10X + 63X - 8+= 180 13X – 2 = 180 +2 13X = 182 13 X= 14 9X + 1020X-7+= 90 29X +3 = 90 -3 29X = 87 29 X = 3 - 14 -14 - X = 8X + 1 -8X -9X = 1 -9 X -.11 14 – X = 8X + 15 9X + 28 = 13X -16 - 28 -28 9X = 13X - 44 -13X -4X= - 44 -4 X = 11 Solve these equations: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 10 - 14 -14 7Y = -12Y +76 + 12Y 19Y = 76 19 Y = 4 7Y + 14 = 90 – 12Y 142 + (2Y + 8) = 180 150 + 2Y = 180 -150 2Y = 30 2 Y = 15 -2Y -2Y 36 = 3Y 3 3 Y=12 2Y+36=5Y 5X + 5 = 7X - 19 -5X -5X 5 = 2X - 19 +19 24 = 2X 2 2 X=12 Solve these equations: Standard 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11 11 100 + (7X) + (10X) + (9X-6) + (10X+3) + (7X-9) = 720 7X + 10X + 9X + 10X + 7X + 100 – 6 +3 -9 = 720 43X + 88 = 720 -88 43 X 14.7 43X = 632 =9037-X +2X-16 +X+13 37 – 16 + 13 –X +2X +X = 90 34 +2X = 90 -34 2X = 56 2 X=28 Solve these equations: Standard 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 12 10 = (10X) 1 2 10 = 5X 5 X = 2 = ( + ) 1 2 3X-1 7X+1 10 (12X+11) + (9X+3) + (7X+26) = 180 28X + 40 = 180 -40 28X = 140 28 X = 5 Solve these equations: 12X + 9X + 7X + 11 + 3 + 26 = 180 2 = X 3(4X – 5)= -2(3X -3) 12X – 15 = -6X +6 +15 12X = -6X + 21 +6X 18X = 21 18 21 18 1 3 X = 1 3 18.... 3 3 = 1 1 6 1 1 6 OR X 1.2 Standard 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 13 Solve these equations: 2 3 = 4(3X-2) 5(2X+6) 2 3 = 12X -8 10X+30 2(10X+30)=3(12X-8) 20X +60 = 36X -24 +24 20X + 84 = 36X -20X 84 = 16X 16 84 16 5 80 4 5 4 16 5 1 4 X = OR X 5.25 3 7 = X+6 X 3X = 7(X+6) 3X = 7X + 42 -7X -4X = 42 -4 42 4 1 4 2 10 2 4.... 2 2 = 1 2 0 OR X= -10.5 Standard 1 X= 10 1 2 -.... 4 4 5 1 4 = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 14 ABSOLUTE VALUE |X|=2 012 3 4 5 6 7 8 9 10 11 12 -2 -3 -4 -5 -6-7 -10 -11-12 -8-9 X= -2 X=2 |X|=12 012 3 4 5 6 7 8 9 10 11 12 -2 -3 -4 -5 -6-7 -10 -11-12 -8-9 X= -12 X=12 For any real number a: If a < 0, then |a|= -a; If a > 0, then |a|= a; Absolute Value: Standard 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 15 X + 4 = 70 -4 X=66 |X + 4|= 70 -(X + 4) = 70 +4 X= -74 - X - 4 = 70 | ( ) + 4 |= 70 66 | ( ) + 4 |= 70 -74 |70 |= 70 70= 70 |-74 + 4 |= 70 |-70 |= 70 70= 70 66 and -74 are solutions Standard 1 |X + 4|= 70 -X = 74 (-1) SOLVE: Check: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 16 10X + 20 = 80 -20 10X = 60 10 X=6 |10X + 20|= 80 -(10X + 20) = 80 +20 -10X = 100 -10 X= -10 -10X - 20 = 80 |10( ) + 20 |= 80 6 -10 |60 + 20 |= 80 |80 |= 80 80= 80 |-100 + 20 |= 80 |-80 |= 80 80= 80 6 and -10 are solutions Standard 1 |10X + 20 |= 80 SOLVE: Check: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17 17 5X + 2 = 52 -2 5X = 50 5 5 X=10 10|5X + 2|= 520 -(5X + 2) = 52 +2 -5X = 54 -5 X= -10.8 -5X - 2 = 52 10|5( ) + 2|= 520 10 10|5( ) + 2 |= 520 -10.8 10|50 + 2 |= 520 10|52 |= 520 520= 520 10|-54 + 2 |= 520 10|-52 |= 80 520= 520 10 and -10.8 are solutions Standard 1 10|5X + 2 |= 520 10 |5X + 2|= 52 10(52)=520 SOLVE: Check: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 18 Standard 1 SOLVE: |6X+10| = 8X+4 - 10 -10 6X = 8X - 6 -8X -8X -2X= - 6 -2 X = 3 +10 -6X = 8X + 14 -8X -14X= 14 -14 X = -1 6X + 10 = 8X + 4 -6X - 10 = 8X + 4 -(6X + 10) = 8X + 4 |6X+10| = 8X+4 |6( )+10| = 8( )+4 3 3 |18+10| = 24 + 4 |28| = 28 28 = 28 |-6+10| =-8+4 |4| = -4 4 = -4 The only solution is 3 Check: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19 19 |8X -9|= -1 Standard 1 No solution because absolute value is never negative. |6X + 7 | + 70 = 20 -70 |6X + 7 | = -50 No solution because absolute value is never negative. SOLVE: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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