Transcript 1.7 Solving Absolute Value Equations & Inequalities

Solving Absolute Value
Equations & Inequalities
Isolate the absolute value
Absolute Value (of x)
 Symbol
lxl
 The distance x is from 0 on the number line.
 Always positive
 Ex: l-3l=3
-4
-3
-2
-1
0
1
2
Ex: x = 5
 What are the possible values of x?
x=5
or
x = -5
To solve an absolute value equation:
ax+b = c, (where c>0)
To solve, set up 2 new equations, then solve each
equation.
ax+b = c
or
ax+b = -c
** make sure the absolute value is by itself before you
split to solve.
Ex: Solve 6x-3 = 15
6x-3 = 15
or
6x-3 = -15
Ex: Solve 6x-3 = 15
6x-3 = 15
or
6x-3 = -15
6x = 18
or
6x = -12
x=3
or
x = -2
* Plug in answers to check your solutions!
Ex: Solve 2x + 7 -3 = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
.
Ex: Solve 2x + 7 -3 = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.
Try this one:
3 x  4  36
Divide by 3
3 x  4  36
x  4  12
x  4  12 or x  4  12
x  16 or x  8
Next example:
2  x  7 16
Solve this example:
2  x  7 16
-2
-2
 x  7  14
x  7  14
x  7  14 , x  7  14
x  7 , x  21
But they do not check
{ }
Last one:
3x  9  5  x
Last one: be sure to negate
the entire other side!
3x  9  5  x
3x  9  x  5
3x  9  x  5 or 3x  9  x  5
Last one:
3x  9  5  x
3x  9  x  5
3x  9  x  5 , 3x  9  x  5
4x = 4
2x  14
x7
X=1
Hand this one in
2x  5 1 10
Answer:
2x  5 1 10
2x  5  11
2x  5  11 , 2x  5  11
2x  6
2x = -16
X = -8
x3
BOTH ANSWERS CHECK.
2  3  5 1  10 , 2  8  5 1  10
16  5 1  10
11 1  10
11 1  10
10  10
11 1  10
11 1  10
10  10
Solving Absolute Value Inequalities
1. ax+b < c, where c>0
Becomes an “and” problem
Changes to: –c<ax+b<c
2. ax+b > c, where c>0
Becomes an “or” problem
Changes to: ax+b>c or ax+b<-c
Interval notation:
 2<x<9
(2,9)
(,3) (2,)
 x<-3 or x > 2

1  x  5
x
[-1,5]
 2 or x  4
(,2][4,)
Note that  and  always have ( ) never [ ]
When it’s less than
 It’s and “and”
4x  9  21
Ex: Solve & graph.
4x  9  21
 Becomes an “and” problem
 21  4 x  9  21
 12  4 x  30
15
3 x 
2
-3
Interval notation:
7
[-3,7.5]
8
Greater than
 Is an or
3x  2  3  11
Solve & graph.
3x  2  3  11
 Get absolute value by itself first.
3x  2  8
 Becomes an “or” problem
3x  2  8 or 3x  2  8
3 x  10 or
3 x  6
10
x
or x  2
3
10 3
(,2][ ,)
3
-2
4
Try this one:
2  x  6  10
Try this one:
2  x  6  10
 x  6  12
x  6  12
Now decide that it is an “and” graph!
answer:
2  x  6  10
 x  6  12
x  6  12
x  6  12
x6
x  6  12
x  18
(18, 6) or  18  x  6