solving inequalities objective: the student will be able to solve inequalities. algebra 2
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Solving Inequalities
Objective: The student will be able to solveinequalities.
Algebra 2
-9 -8 -7 -6 -5 -4 -3 -2
Inequalities -2x - 6 > 4
-2x > 10 -2 -2
x < -5
+6 +6
E
Solve and Graph the Following
6. -x - 2 > 67. 5x - 6 ≥ 2x8. -7 < 2x - 3 < 3
-12 -11 -10 -9 -8 -7 -6 -5
Inequalities -x - 2 > 6
-x > 8 -1 -1
x < -8
+2 +2
6
-2 -1 0 1 2 3 4 5
Inequalities 5x - 6 ≥ 2x
3x - 6 ≥ 0
3 3x ≥ 2
-2x -2x
+6 +6 3x ≥ 6
OR
7
-2 -1 0 1 2 3 4 5
Inequalities 5x - 6 ≥ 2x
- 6 ≥ -3x -3 -3
2 ≤ x
-5x -5x
x ≥ 2
7
-3 -2 -1 0 1 2 3 4
Inequalities -7 < 2x - 3 < 3
-4 < 2x < 62 2 2
-2 < x < 3
+3 +3 +3
8
Solve and Graph the Following
9. -11 ≤ 3x - 5 ≤ -210. -19 < 4x - 3 ≤ 1
-4 -3 -2 -1 0 1 2 3
Inequalities -11 ≤ 3x - 5 ≤ -2
-6 ≤ 3x ≤ 33 3 3
-2 ≤ x ≤ 1
+5 +5 +5
9
-5 -4 -3 -2 -1 0 1 2
Inequalities -19 < 4x - 3 ≤ 1
-16 < 4x ≤ 44 4 4
-4 < x ≤ 1
+3 +3 +3
10
andversus
or
Inequalities E
-4 -3 -2 -1 0 1 2 3
x < 2 and x > -2
Inequalities
{ }x-2 2< <
E
-2 -1 0 1 2 3 4 5
and x > -1
{ } x > 1
Inequalities x > 1
E
-5 -4 -3 -2 -1 0 1 2
no solutions
andx < -2 x > -1
Inequalities E
-3 -2 -1 0 1 2 3 4
all real numbers
orall realnumbers x < 1
Inequalities E
-4 -3 -2 -1 0 1 2 3
-1 -1x + 1 < 3 and x - 2 > -4
x < 2 +2 +2
and x > -2
Inequalities
{ }x-2 2< <
11
-3 -2 -1 0 1 2 3 4
+1 +1x - 1 < 2 and x - 1 > -2
x < 3 +1 +1
and x > -1
Inequalities
{ }x-1 3< <
12
-x -xx - 2 < 2x and 3x - 2 > 1
-2 < x +2 +2
3x > 3and 3-3 x > -2
{ } x > 1
Inequalities
x > 1
13
-3 -2 -1 0 1 2 3 4
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
|7| = 7
|-7| = 7
Absolute Value Equations
|x| = 4x = 4 or x = -4
|x| = 5x = 5 or x = -5
Absolute Value Equations
|x| = - 4No Solutions
|x+2| = 7x+2 = 7 or x+2 = -7
Absolute Value Equations
|x-8| = xx-8 = x or x-8 = -x
|2x+2| = x2x+2 = x or 2x+2 = -x
Absolute Value Equations
|x-4| = x + 2x-4 = x+2 or x-4 = -x-2
|x+5| = 3x-4x+5 = 3x-4 or x+5 = -3x+4
|x - 1| + 5 = 7 -5 -5
|x - 1| = 2
Absolute Value EquationsE
Absolute Value Equations |x - 1| = 2
+1 +1 x - 1 = 2 or x - 1 = -2
x = 3+1 +1
x = -1orAnswers must be checked!
E
|x - 1| + 5 = 7
|(3) - 1| + 5 = 7
Absolute Value EquationsE
|2| + 5 = 72 + 5 = 7
7 = 7
|x - 1| + 5 = 7
|(-1) - 1| + 5 = 7
Absolute Value EquationsE
|-2| + 5 = 72 + 5 = 7
7 = 7
Absolute Value Equations |x - 1| = 2
+1 +1 x - 1 = 2 or x - 1 = -2
x = 3+1 +1
x = -1or
{3, -1}
E
Absolute Value Equations |x - 2| = 4
+2 +2 x - 2 = 4 or x - 2 = -4
x = 6+2 +2
x = -2orAnswers must be checked!
14
|x - 2| = 4
|(6) - 2| = 4
|4| = 4
4 = 4
Absolute Value Equations14
|x - 2| = 4
|(-2) - 2| = 4
|-4| = 4
4 = 4
Absolute Value Equations14
Absolute Value Equations |x - 2| = 4
+2 +2 x - 2 = 4 or x - 2 = -4
x = 6+2 +2
x = -2or
{6, -2}
14
Absolute Value EquationsE|2x - 6| = 4x
-2x -2x2x - 6 = 4x or 2x - 6 = -4x
-6 = 2x 2 2-3 = x
Absolute Value EquationsE|2x - 6| = 4x
|2(-3) - 6| = 4(-3)
|-6 - 6| = -12
12 = -12|-12| = -12
Absolute Value EquationsE|2x - 6| = 4x
-2x -2x2x - 6 = 4x or 2x - 6 = -4x
-6 = 2x-2x -2x
-6 = -6x
or 2 2-3 = x 1 = x
-6 -6
Absolute Value EquationsE|2x - 6| = 4x
|2(1) - 6| = 4(1)
|2 - 6| = 4
4 = 4 |-4| = 4
Absolute Value EquationsE|2x - 6| = 4x
-2x -2x2x - 6 = 4x or 2x - 6 = -4x
-6 = 2x-2x -2x
-6 = -6x
or 2 2-3 = x 1 = x
-6 -6
{1}
|x - 2| = 2x
-x -xx - 2 = 2x or x - 2 = -2x
-2 = x
Absolute Value Equations15
Absolute Value Equations15
|(-2) - 2| = 2(-2)
|-4| = -4
4 = -4
|x - 2| = 2x
-x -xx - 2 = 2x or x - 2 = -2x
-2 = x-x -x
-2 = -3xor-3-3
= x23
Absolute Value Equations15|x - 2| = 2x
Absolute Value Equations15
|( ) - 2| = 2( )
|- | =
=
23
23
43
4343
43
|x - 2| = 2x
-x -xx - 2 = 2x or x - 2 = -2x
-2 = x-x -x
-2 = -3xor-3-3
= x23
Absolute Value Equations15
{ }23
|x - 2| = 2x
|4x - 6| = 2x
-4x -4x4x - 6 = 2x or 4x - 6 = -2x
-6 = -2x-2 -23 = x
Absolute Value Equations16
|4x - 6| = 2x
|4(3) - 6| = 2(3)
|12 - 6| = 6
|6| = 6
6 = 6
Absolute Value Equations16
|4x - 6| = 2x
-4x -4x4x - 6 = 2x or 4x - 6 = -2x
-6 = -2x-4x -4x
-6 = -6x
or-2 -23 = x 1 = x
-6 -6
Absolute Value Equations16
|4x - 6| = 2x
|4(1) - 6| = 2(1)
|4 - 6| = 2
|-2| = 2
2 = 2
Absolute Value Equations16
|4x - 6| = 2x
-4x -4x4x - 6 = 2x or 4x - 6 = -2x
-6 = -2x-4x -4x
-6 = -6x
or-2 -23 = x 1 = x
-6 -6
{1, 3}
Absolute Value Equations16
Absolute Value Inequalities
Objective: The student will be able to solve absolute value inequalities.
Algebra 2
Absolute Value Inequalities
|x| ≥ 2
-4 -3 -2 -1 0 1 2 3
-4 -3 -2 -1 0 1 2 3
Absolute Value Inequalities
|x| ≥ 2
-4 -3 -2 -1 0 1 2 3
x ≥ 2orx ≤ -2
|x| ≥ 1x ≥ 1 or x ≤ -1
-4 -3 -2 -1 0 1 2 3
Absolute Value Inequalities
Absolute Value Inequalities
|x| ≤ 2
-4 -3 -2 -1 0 1 2 3
-4 -3 -2 -1 0 1 2 3
x ≤ 2 and x ≥ -2
|x| ≤ 2
x ≤ 2 and x ≥ -2
-4 -3 -2 -1 0 1 2 3
Absolute Value Inequalities
{ }x-2 2≤ ≤
|x| ≤ 3
x ≤ 3 and x ≥ -3
-4 -3 -2 -1 0 1 2 3
Absolute Value Inequalities
{ }x-3 3≤ ≤
|x + 4| > 2x - 5
Write the compound inequalitythat 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 - 5 or x + 4 < -2x + 5
Write the compound inequalitythat is equivalent to the given.
Absolute Value Inequalities
|-2x + 4| ≤ x
-2x + 4 ≤ x and -2x + 4 ≥ -x
Write the compound inequalitythat is equivalent to the given.
Absolute Value Inequalities
|x - 3| < 2x
-x -x-3 < x
-x -x
and-3 > -3x
Absolute Value InequalitiesE1
x > -3 -3-31 < xx > 1
x - 3 < 2x and x - 3 > -2x
-3 -2 -1 0 1 2 3 4
Absolute Value InequalitiesE1
andx > -3 x > 1
{x > 1}
|4x| < 2x - 6
-2x -2x2x < -6
+2x +2x6x > 6
and6622
x < -3 x > 1
4x < 2x - 6 and 4x > -2x + 6
Absolute Value Inequalities17
|4x| < 2x - 6
-5 -4 -3 -2 -1 0 1 2
no solutions
andx < -3 x > 1
Absolute Value Inequalities17
|2x| > 2x - 4
-2x -2x0 > -4
+2x +2x4x < 4
or 44all realnumbers x < 1
2x > 2x - 4 or 2x < -2x + 4
Absolute Value Inequalities18
|2x| > 2x - 4
-3 -2 -1 0 1 2 3 4
all real numbers
orall realnumbers x < 1
Absolute Value Inequalities18
Absolute Value Inequalities
|x| ≥ 2
-4 -3 -2 -1 0 1 2 3
-4 -3 -2 -1 0 1 2 3
Absolute Value Inequalities
|x| ≥ 2
-4 -3 -2 -1 0 1 2 3
x ≥ 2orx ≤ -2
|x| ≥ 1x ≥ 1 or x ≤ -1
-4 -3 -2 -1 0 1 2 3
Absolute Value Inequalities
Absolute Value Inequalities
|x| ≤ 2
-4 -3 -2 -1 0 1 2 3
-4 -3 -2 -1 0 1 2 3
x ≤ 2 and x ≥ -2
|x| ≤ 2
x ≤ 2 and x ≥ -2
-4 -3 -2 -1 0 1 2 3
Absolute Value Inequalities
{ }x-2 2≤ ≤
|x| ≤ 3
x ≤ 3 and x ≥ -3
-4 -3 -2 -1 0 1 2 3
Absolute Value Inequalities
{ }x-3 3≤ ≤
|x + 4| > 2x - 5
x + 4 > 2x - 5 or x + 4 < -2x + 5
Absolute Value Inequalities
Write the compound inequalitythat is equivalent to the given.
|-2x + 4| ≤ x
-2x + 4 ≤ x and -2x + 4 ≥ -x
Absolute Value Inequalities
Write the compound inequalitythat is equivalent to the given.
|x - 3| < 2x
-x -x-3 < x
-x -x
and-3 > -3x
Absolute Value InequalitiesE1
x > -3 -3-31 < xx > 1
x - 3 < 2x and x - 3 > -2x
-3 -2 -1 0 1 2 3 4
Absolute Value InequalitiesE1
andx > -3 x > 1
{x > 1}
|4x| < 2x - 6
-2x -2x2x < -6
+2x +2x6x > 6
and6622
x < -3 x > 1
4x < 2x - 6 and 4x > -2x + 6
Absolute Value Inequalities19
|4x| < 2x - 6
-5 -4 -3 -2 -1 0 1 2
no solutions
andx < -3 x > 1
Absolute Value Inequalities19
|2x| > 2x - 4
-2x -2x0 > -4
+2x +2x4x < 4
or 44all realnumbers x < 1
2x > 2x - 4 or 2x < -2x + 4
Absolute Value Inequalities20
|2x| > 2x - 4
-3 -2 -1 0 1 2 3 4
all real numbers
orall realnumbers x < 1
Absolute Value Inequalities20