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 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 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 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 = = = = 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 = = = = 4 24 = = = = 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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

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

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

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

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

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

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

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 X + 84 = 36X -20X 84 = 16X X = OR X = X+6 X 3X = 7(X+6) 3X = 7X X -4X = = OR X= Standard 1 X= = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 ABSOLUTE VALUE |X|= X= -2 X=2 |X|= 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 X + 4 = X=66 |X + 4|= 70 -(X + 4) = X= X - 4 = 70 | ( ) + 4 |= | ( ) + 4 |= |70 |= 70 70= 70 | |= 70 |-70 |= 70 70= and -74 are solutions Standard 1 |X + 4|= 70 -X = 74 (-1) SOLVE: Check: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 10X + 20 = X = X=6 |10X + 20|= 80 -(10X + 20) = X = X= X - 20 = 80 |10( ) + 20 |= | |= 80 |80 |= 80 80= 80 | |= 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 5X + 2 = X = X=10 10|5X + 2|= 520 -(5X + 2) = X = X= X - 2 = 52 10|5( ) + 2|= |5( ) + 2 |= | |= |52 |= = | |= |-52 |= = and are solutions Standard 1 10|5X + 2 |= |5X + 2|= 52 10(52)=520 SOLVE: Check: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 Standard 1 SOLVE: |6X+10| = 8X X = 8X X -8X -2X= X = X = 8X X -14X= X = -1 6X + 10 = 8X X - 10 = 8X + 4 -(6X + 10) = 8X + 4 |6X+10| = 8X+4 |6( )+10| = 8( ) |18+10| = |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 |8X -9|= -1 Standard 1 No solution because absolute value is never negative. |6X + 7 | + 70 = |6X + 7 | = -50 No solution because absolute value is never negative. SOLVE: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved