Solving Inequalities

Objective: The student will be able to solve inequalities. Algebra 2

Inequalities -2x - 6 > 4 -2x > x < E

Solve and Graph the Following 6.-x - 2 > 6 7.5x - 6 ≥ 2x 8.-7 < 2x - 3 < 3

Inequalities -x - 2 > 6 -x > x <

Inequalities 5x - 6 ≥ 2x 3x - 6 ≥ x ≥ 2 -2x -2x x ≥ 6 OR 7

Inequalities 5x - 6 ≥ 2x - 6 ≥ -3x ≤ x -5x -5x x ≥ 2 7

Inequalities -7 < 2x - 3 < 3 -4 < 2x < < x <

Solve and Graph the Following ≤ 3x - 5 ≤ < 4x - 3 ≤ 1

Inequalities -11 ≤ 3x - 5 ≤ ≤ 3x ≤ ≤ x ≤

Inequalities -19 < 4x - 3 ≤ < 4x ≤ < x ≤

and versus or Inequalities E

x < 2 and x > -2 Inequalities { }x-2 2< < E

and x > -1 { } x > 1 Inequalities x > 1 E

no solutions and x < -2 x > -1 Inequalities E

all real numbers or all real numbers x < 1 Inequalities E

x x < and x > -2 Inequalities { }x-2 2< < 11

x x < and x > -1 Inequalities { }x 3< < 12

-x x < x x > 3 and 3-3 x > -2 { } x > 1 Inequalities x >

Absolute Value Equations

Objective: The student will be able to solve absolute value equations. Algebra 2 SOL A2.4

Absolute Value |6| =|6| = 6 |-6| =|-6| = 6 |7| =|7| = 7 |-7| =|-7| = 7

Absolute Value Equations | x | = 4 x = 4 or x = -4 | x | = 5 x = 5 or x = -5

Absolute Value Equations | x | = - 4 No Solutions | x+2 | = 7 x+2 = 7 or x+2 = -7

Absolute Value Equations | x-8 | = x x-8 = x or x-8 = -x | 2x+2 | = x 2x+2 = x or 2x+2 = -x

Absolute Value Equations | x-4 | = x + 2 x-4 = x+2 or x-4 = -x-2 | x+5 | = 3x-4 x+5 = 3x-4 or x+5 = -3x+4

| x - 1 | + 5 = | x - 1 | = 2 Absolute Value Equations E

| x - 1 | = x - 1 = 2 or x - 1 = -2 x = 3 +1 x = -1 or Answers must be checked! E

| x - 1 | + 5 = 7 | (3) - 1 | + 5 = 7 Absolute Value Equations E | 2 | + 5 = = 7 7 = 7

| x - 1 | + 5 = 7 | (-1) - 1 | + 5 = 7 Absolute Value Equations E | -2 | + 5 = = 7 7 = 7

Absolute Value Equations | x - 1 | = x - 1 = 2 or x - 1 = -2 x = 3 +1 x = -1 or {3, -1} E

Absolute Value Equations | x - 2 | = x - 2 = 4 or x - 2 = -4 x = 6 +2 x = -2 or Answers must be checked! 14

| x - 2 | = 4 | (6) - 2 | = 4 | 4 | = 4 4 = 4 Absolute Value Equations 14

| x - 2 | = 4 | (-2) - 2 | = 4 | -4 | = 4 4 = 4 Absolute Value Equations 14

Absolute Value Equations | x - 2 | = x - 2 = 4 or x - 2 = -4 x = 6 +2 x = -2 or {6, -2} 14

Absolute Value Equations E | 2x - 6 | = 4x -2x 2x - 6 = 4x or 2x - 6 = -4x -6 = 2x = x

Absolute Value Equations E | 2x - 6 | = 4x | 2(-3) - 6 | = 4(-3) | | = = -12  | -12 | = -12

Absolute Value Equations E | 2x - 6 | = 4x -2x 2x - 6 = 4x or 2x - 6 = -4x -6 = 2x -2x -6 = -6x or = x 1 = x -6

Absolute Value Equations E | 2x - 6 | = 4x | 2(1) - 6 | = 4(1) | | = 4 4 = 4 | -4 | = 4

Absolute Value Equations E | 2x - 6 | = 4x -2x 2x - 6 = 4x or 2x - 6 = -4x -6 = 2x -2x -6 = -6x or = x 1 = x -6 {1}

| x - 2 | = 2x -x x - 2 = 2x or x - 2 = -2x -2 = x Absolute Value Equations 15

Absolute Value Equations 15 | (-2) - 2 | = 2(-2) | -4 | = -4 4 = -4  | x - 2 | = 2x

-x x - 2 = 2x or x - 2 = -2x -2 = x -x -2 = -3x or -3 = x 2 3 Absolute Value Equations 15 | x - 2 | = 2x

Absolute Value Equations 15 | ( ) - 2 | = 2( ) | - | = = | x - 2 | = 2x

-x x - 2 = 2x or x - 2 = -2x -2 = x -x -2 = -3x or -3 = x 2 3 Absolute Value Equations 15 { } 2323 | x - 2 | = 2x

| 4x - 6 | = 2x -4x 4x - 6 = 2x or 4x - 6 = -2x -6 = -2x -2 3 = x Absolute Value Equations 16

| 4x - 6 | = 2x | 4(3) - 6 | = 2(3) | | = 6 | 6 | = 6 6 = 6 Absolute Value Equations 16

| 4x - 6 | = 2x -4x 4x - 6 = 2x or 4x - 6 = -2x -6 = -2x -4x -6 = -6x or -2 3 = x 1 = x -6 Absolute Value Equations 16

| 4x - 6 | = 2x | 4(1) - 6 | = 2(1) | | = 2 | -2 | = 2 2 = 2 Absolute Value Equations 16

| 4x - 6 | = 2x -4x 4x - 6 = 2x or 4x - 6 = -2x -6 = -2x -4x -6 = -6x or -2 3 = x 1 = x -6 {1, 3} Absolute Value Equations 16

Absolute Value Inequalities

Objective: The student will be able to solve absolute value inequalities. Algebra 2

Absolute Value Inequalities | x | ≥

Absolute Value Inequalities | x | ≥ x ≥ 2orx ≤ -2

| x | ≥ 1 x ≥ 1 or x ≤ Absolute Value Inequalities

| x | ≤ x ≤ 2 and x ≥ -2

| x | ≤ 2 x ≤ 2 and x ≥ Absolute Value Inequalities { }x-22≤≤

| x | ≤ 3 x ≤ 3 and x ≥ Absolute Value Inequalities { }x-33≤≤

| x + 4 | > 2x - 5 Write the compound inequality that is equivalent to the given. | -2x + 4 | ≤ x Absolute Value Inequalities

< Less Thand > Greator> Greater < Less Than Absolute Value Inequalities

| x + 4 | > 2x - 5 x + 4 > 2x - 5orx + 4<-2x+ 5 Write the compound inequality that is equivalent to the given. Absolute Value Inequalities

| -2x + 4 | ≤ x -2x + 4 ≤ x and -2x + 4 ≥-x Write the compound inequality that is equivalent to the given. Absolute Value Inequalities

|x - 3| < 2x -x -3 < x -x -x and -3 > -3x Absolute Value Inequalities E1 x > < x x > 1 x - 3 < 2xandx - 3>-2x

Absolute Value Inequalities E1 and x > -3 x > 1 {x > 1}

|4x| < 2x x 2x < -6 +2x +2x 6x > 6 and x < -3 x > 1 4x < 2x - 6and4x>-2x+ 6 Absolute Value Inequalities 17

|4x| < 2x no solutions and x < -3 x > 1 Absolute Value Inequalities 17

|2x| > 2x x 0 > -4 +2x +2x 4x < 4 or 4 4 all real numbers x < 1 2x > 2x - 4or2x<-2x+ 4 Absolute Value Inequalities 18

|2x| > 2x all real numbers or all real numbers x < 1 Absolute Value Inequalities 18

Absolute Value Inequalities | x | ≥

Absolute Value Inequalities | x | ≥ x ≥ 2orx ≤ -2

| x | ≥ 1 x ≥ 1 or x ≤ Absolute Value Inequalities

| x | ≤ x ≤ 2 and x ≥ -2

| x | ≤ 2 x ≤ 2 and x ≥ Absolute Value Inequalities { }x-22≤≤

| x | ≤ 3 x ≤ 3 and x ≥ Absolute Value Inequalities { }x-33≤≤

| x + 4 | > 2x - 5 x + 4 > 2x - 5orx + 4<-2x+ 5 Absolute Value Inequalities Write the compound inequality that is equivalent to the given.

| -2x + 4 | ≤ x -2x + 4 ≤ x and -2x + 4 ≥-x Absolute Value Inequalities Write the compound inequality that is equivalent to the given.

|x - 3| < 2x -x -3 < x -x -x and -3 > -3x Absolute Value Inequalities E1 x > < x x > 1 x - 3 < 2xandx - 3>-2x

Absolute Value Inequalities E1 and x > -3 x > 1 {x > 1}

|4x| < 2x x 2x < -6 +2x +2x 6x > 6 and x < -3 x > 1 4x < 2x - 6and4x>-2x+ 6 Absolute Value Inequalities 19

|4x| < 2x no solutions and x < -3 x > 1 Absolute Value Inequalities 19

|2x| > 2x x 0 > -4 +2x +2x 4x < 4 or 4 4 all real numbers x < 1 2x > 2x - 4or2x<-2x+ 4 Absolute Value Inequalities 20

|2x| > 2x all real numbers or all real numbers x < 1 Absolute Value Inequalities 20