SOLUTION OF INEQUALITIES INCLUDING ABSOLUTE VALUE INEQUALITIES Standard 1 SOLUTION OF INEQUALITIES INCLUDING ABSOLUTE VALUE INEQUALITIES END SHOW PROPERTIES OF INEQUALITY INEQUALITIES LEVEL 1 INEQUALITIES LEVEL 2 INEQUALITIES LEVEL 3 INEQUALITIES LEVEL 4 COMPOUND INEQUALITIES “OR” COMPOUND INEQUALITIES “AND” ABSOLUTE VALUE INEQUALITIES “AND” ABSOLUTE VALUE INEQUALITIES “OR” PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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

PROPERTIES OF INEQUALITY Standard 1 PROPERTIES OF INEQUALITY ADDITION AND SUBTRACTION PROPERTIES OF INEQUALITY: For any numbers a, b, and c, if a>b then a+c>b+c and a-c>b-c 20 > 10 40 > 30 + 6 +6 - 7 -7 26 > 16 33 > 23 For any numbers a, b, and c, if a<b then a+c<b+c and a-c<b-c 12 < 22 34 < 50 +5 +5 -2 -2 17 < 27 32 < 48 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF INEQUALITY Standard 1 PROPERTIES OF INEQUALITY MULTIPLICATION AND DIVISION PROPERTIES OF INEQUALITY If c>0 AND a > b then: a c b ac>bc and > 42 > 28 2 18 > 15 7 7 36 > 30 6 > 4 36 > 12 3 40 > 24 12 12 120> 72 3 > 1 If c>0 AND a < b then: a c b ac<bc and < 4 22 < 30 55 < 110 11 11 88 < 120 5 < 10 40 < 70 5 10 < 20 10 10 50< 100 4 < 7 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x X X F PROPERTIES OF INEQUALITY Standard 1 PROPERTIES OF INEQUALITY MULTIPLICATION AND DIVISION PROPERTIES OF INEQUALITY If c<0 AND a > b then: a c b ac<bc and < 42 > 28 -2 18 > 15 -7 -7 -36 < -30 -6 < -4 25 50 X -25 -50 x 4 2 6 -2 -4 -6 8 10 -8 -10 36 > 12 -3 40 > 24 -12 -12 -120 < -72 -3 < -1 100 200 X -100 -200 F -2 -4 -6 -8 -10 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

X F F X PROPERTIES OF INEQUALITY Standard 1 PROPERTIES OF INEQUALITY MULTIPLICATION AND DIVISION PROPERTIES OF INEQUALITY If c<0 AND a < b then: a c b ac>bc and > -4 22 < 30 55 < 110 -11 -11 -88 > -120 -5 > -10 100 200 X -100 -200 F -2 -4 -6 -8 -10 40 < 70 -5 10 < 20 -10 -10 -50> -100 -4 > -7 100 200 X -100 -200 F -2 -4 -6 -8 -10 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x K x F x Solve these inequalities then graph the solution: Standard 1 Solve these inequalities then graph the solution: 25 > 12 + K X + 4 < 12 -4 -4 -12 -12 X < 8 13 > K X < 8 X K <13 K < 13 K x 4 2 6 -2 -4 -6 8 10 -8 -10 5 10 15 20 25 K -5 -10 -15 -20 6X < -60 10X 20 < -9F 54 > 10 10 -9 -9 X 2 < F -6 < 6 6 X 2 X < X < -10 F -6 F < x 4 2 6 -2 -4 -6 8 10 -8 -10 F -2 -4 -6 -8 -10 X< -10 X x 4 2 6 -2 -4 -6 8 10 -8 -10 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x X Z X Solve these inequalities and graph the solution: X 7 X Standard 1 Solve these inequalities and graph the solution: 20X + 40 180 > -24 > 8 -16X -40 -40 -8 -8 -32 > -16X 20X 140 > -16 -16 20 20 2 < X X 7 > X > 2 X 7 X > X > 2 X x 4 2 6 -2 -4 -6 8 10 -8 -10 X 4 2 6 -2 -4 -6 8 10 -8 -10 32 < 3X- 1 10 + 4Z 26 < +1 +1 -10 -10 33< 3X 4Z 16 < 3 3 4 4 X>11 Z 4 < X >11 X Z 4 Z < 5 10 15 20 25 X -5 -10 -15 -20 Z 4 2 6 -2 -4 -6 8 10 -8 -10

Y X x R Solve these inequalities and graph the solution: + 2 < 34 X Standard 1 Solve these inequalities and graph the solution: + 2 < 34 X 4 10- 46 Y 8 > -2 -2 -10 -10 X 4 <32 (4) (4) -Y 8 36 > (-8) (-8) X < 128 X < 128 X Y -288 < Y -288 Y < -288 128 Y -100 -200 -300 100 200 X -100 -200 R< 2 2 3 X – 1 2 7 > +1 +1 3 2 R< 2 2 3 7 3 X 3 3 7 > R < 3 R R < 3 X 7 > X 7 X > x 4 2 6 -2 -4 -6 8 10 -8 -10 R 4 2 6 -2 -4 -6 8 10 -8 -10

X X Solve these inequalities and graph the solution: 12 -4X 8 8 > Standard 1 Solve these inequalities and graph the solution: 12 -4X 8 8 > 10X - 6 4 <12 12-4X 8 8 > 10X - 6 4 <12 (8) (8) (4) (4) 64 > 12– 4X 10X – 6< 48 -12 -12 +6 +6 52 > -4X 5 10X < 54 . 2 54 10 5 4 10 = 5 2 -4 -4 10 10 50 -13 < X X < 5 2 4 X > -13 X > -13 X 5 2 X < X 5 10 15 20 25 X -5 -10 -15 -20 X 4 2 6 -2 -4 -6 8 10 -8 -10 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standard 1 Solve this inequality and graph it: 20X +12 6X-16 + 360 > 26X – 4 360 > +4 +4 26X 364 > 26 26 X 14 > X 14 X > 5 10 15 20 25 X -5 -10 -15 -20 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Y Y Solve these inequalities and graph them: Y Y > 15 Y Y < 4 Standard 1 Solve these inequalities and graph them: 14Y + 28 < 180 – 24Y 71 + ( Y + 4) > 90 - 28 -28 75 + Y > 90 14Y < -24Y +152 - 75 -75 + 24Y + 24Y Y > 15 38Y < 152 38 38 Y > 15 Y Y < 4 5 10 15 20 25 Y -5 -10 -15 -20 Y < 4 Y Y 4 2 6 -2 -4 -6 8 10 -8 -10 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

X Solve this inequality and graph it: 2X+12 3 > 7 2X . = X X < Standard 1 Solve this inequality and graph it: 3 7 > 2X+12 2X 6X > 7(2X+12) 6X > 14X + 84 1 -14X -14X . 4 84 8 10 4 8 = 10 1 2 -8X > 84 8 -8 -8 04 X< 10 1 2 - X < X 10 1 2 - 5 10 15 20 25 X -5 -10 -15 -20 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6+5 < X 17+X < 9 9+X 16 11 < X 9+X 16 17+X < 9 -9 -9 Standard 1 Solve these compound inequalities: 6+5 < X 17+X < 9 OR 9+X 16 > OR 11 < X 9+X 16 > 17+X < 9 -9 -9 X > 11 -17 -17 X < -8 X 7 > -8 7 x 7 x -8 x 11 5 10 15 20 25 x -5 -10 -15 -20 X | OR X<-8 X 7 > PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10+2 < X 21+X < 17 11+X 15 12 < X 11+X 15 21+X < 17 Standard 1 Solve these compound inequalities: 10+2 < X 21+X < 17 OR 11+X 15 > OR 12 < X 11+X 15 > 21+X < 17 -11 -11 X > 12 -21 -21 X < -4 X 4 > -4 4 x 4 x -4 x 12 5 10 15 20 25 x -5 -10 -15 -20 X | OR X<-4 X 4 > PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11+3 < X 15+X > 19 16+X 18 14 < X 16+X 18 15+X > 19 Standard 1 Solve these compound inequalities: 11+3 < X 15+X > 19 OR 16+X 18 > OR 14 < X 16+X 18 > 15+X > 19 -16 -16 X > 14 -15 -15 X > 4 X 2 > 2 x 2 x 4 x 14 5 10 15 20 25 x -5 -10 -15 -20 X | X 2 > PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11+3 >X 15+X > 19 16+X 18 14 > X 16+X 18 15+X > 19 -16 -16 Standard 1 Solve these compound inequalities: 11+3 >X 15+X > 19 OR OR 16+X 18 > 14 > X 16+X 18 > 15+X > 19 -16 -16 X < 14 -15 -15 X > 4 X 2 > x 2 x 4 x 14 5 10 15 20 25 x -5 -10 -15 -20 X | R PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15+8 > X 15+X > 8 8+X > 15 23 > X 8+X > 15 15+X > 8 Standard 1 Solve these compound inequalities: 15+8 > X 15+X > 8 8+X > 15 AND AND 23 > X 8+X > 15 15+X > 8 -8 -8 X < 23 -15 -15 X > -7 X > 7 x 7 x -7 x 23 5 10 15 20 25 x -5 -10 -15 -20 7 23 X | 7<X<23 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

(X+4)+(3X-5) > X (X+4 )+X > (3X-5) (3X-5) +X > (X+4 ) Standard 1 Solve these compound inequalities: (X+4)+(3X-5) > X (X+4 )+X > (3X-5) AND AND (3X-5) +X > (X+4 ) 4X -1 >X 2X +4 > 3X-5 4X – 5 > X +4 -4X -4X -2X -2X -X -X 4 > X-5 3X – 5 > 4 -1 >-3X +5 +5 -3 -3 +5 +5 3X > 9 .33 <X 9 > X 3 3 X < 9 X>.33 X > 3 x 3 x 9 x .33 5 10 15 20 25 x -5 -10 -15 -20 3 9 X | 3<X< 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6+5 < X 17+X < 9 9+X 16 11 < X 9+X 16 17+X < 9 -9 -9 Standard 1 Solve these compound inequalities: 6+5 < X 17+X < 9 AND AND 9+X 16 > 11 < X 9+X 16 > 17+X < 9 -9 -9 X > 11 -17 -17 X < -8 X 7 > x 7 x -8 x 11 5 10 15 20 25 x -5 -10 -15 -20 NO SOLUTION! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x x x X | -4<X<4 Standard 1 ABSOLUTE VALUE INEQUALITIES |X|<4 2 4 6 8 10 x -2 -4 -6 -8 X | -4<X<4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x x X X | -76<X< 64 Standard 1 SOLVE: |X + 6|< 70 -6 -6 -6 -6 X<64 X > -76 x 64 -76 64 x -76 100 200 X -100 -200 X | -76<X< 64 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x x X X | -10<X< 6 Standard 1 SOLVE: |5X + 10|<40 -10 -10 -10 -10 5X > -50 5X < 30 5 5 5 5 X > -10 X < 6 x 6 -10 6 x -10 10 20 X -10 -20 X | -10<X< 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x x X X | -5.4<X< 5 Standard 1 20|10X + 2|< 1040 SOLVE: 20 -2 -2 -2 -2 10X > -54 10X < 50 10 10 10 10 X > -5.4 X < 5 x 5 -54 5 x -5.4 5 10 X -5 -10 X | -5.4<X< 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x x x X | X>3 Standard 1 SOLVE: |12X+20| < 16X+8 - 20 -20 12X + 20 > -16X - 8 12X < 16X - 12 -20 -20 -16X -16X 12X > -16X - 28 -4X< - 12 +16X +16X -4 -4 28X > -28 X > 3 28 28 X > -1 3 x -1 x 3 2 4 6 8 10 x -2 -4 -6 -8 X | X>3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x x x X | OR X<-6 X>6 Standard 1 ABSOLUTE VALUE INEQUALITIES 2 4 6 8 10 x -2 -4 -6 -8 X | OR X<-6 X>6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x x X X | OR X<-55 X>45 Standard 1 SOLVE: |X + 5|> 50 -5 -5 -5 -5 X > 45 X < -55 x 45 -55 45 x -55 100 200 X -100 -200 X | OR X<-55 X>45 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x x X X | OR X<-10 X>6 Standard 1 SOLVE: |15X + 30|>120 -30 -30 -30 -30 15X < -150 15X > 90 15 15 15 15 X < -10 X > 6 x 6 -10 6 x -10 20 40 X -20 -40 X | OR X<-10 X>6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standard 1 |5X -1|> -3 |2X + 7 | + 76 < 20 |2X + 7 | < -56 SPECIAL CASES TO PAY ATTENTION! SOLVE: |5X -1|> -3 Solution are all reals because absolute value is always positive and greater than any negative value. SOLVE: |2X + 7 | + 76 < 20 -76 -76 |2X + 7 | < -56 No solution because absolute value is always positive and never less than a negative value. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved