Download of x will make the expression equal to

Absolute Values
Review
Defn: A non-directional distance from zero.
Absolute values are always non-negative.
x 4
x  4 and ,  4
x  21
x  21 and ,  21
x  7
Does not exist.
Absolute Values
Absolute Values of Expression
What value(s) of x will make the
expression equal to 8 or -8?
3x  4  8
3x  4  8
3x  4  8
3x  12
3x  4
4
x
3
x4
4
solution : x   , 4
3
Absolute Values
Absolute Values of Expressions
What value(s) of x will make the
expression equal to 7 or -7?
 5x  3  7
 5x  3  7
 5x  3  7
 5x  4
 5x  10
4
x
5
x2
4
solution : x   , 2
5
Absolute Values
Absolute Values of Expressions
2x  3  0
What value(s) of x will make the
expression equal to 0?
2x  3  0
2x  3
3
x
2
Absolute Values
Absolute Values of Expressions
What value(s) of x will make the
expression equal to -1?
7
5
6x  6  1  6x  6  1 6   6  1 6   6  1
6
6
6 x  6  1
6x  5
 6x  6  1
5
x
6
 6x  7
5  6  1
7  6  1
7
x
6
1  1
1  1
1  1
1  1
No solution
No solution
Absolute Values
Absolute Values of Inequalities
3x  4  8
What value(s) of x will make the
inequality greater than or equal to 8?
3x  4  8
 3x  4  8
3x  12
3x  4  8
x4
3x  4
4
x
3
Interval Notation
4

  ,   4,  
3

Absolute Values
Absolute Values of Inequalities
3x  4  8
What value(s) of x will make the
inequality greater than or equal to 8?
3x  4  8
 3x  4  8
3x  12
3x  4  8
x4
3x  4
4
x
3
Interval Notation
 4 
  , 4
 3 
Absolute Values
Absolute Values of Other Inequalities
2 x 3  4  2
x 3  3
 x  3  3
2 x 3  6
x6
x  3  3
x0
x 3  3
Interval Notation
 , 0  6, 
Absolute Values
Absolute Values of Other Inequalities
2x  3  x  5
2x  3  x  5
2 x  3  x  5
2x  x  8
2x  3   x  5
x 8
2x   x  2
3x  2
2
x
3
Graphs, Relations, and Functions
Defn: A relation is a set of ordered pairs.
A  0,0, 1,1, 1,1, 4,2, 4,2
Domain: The values of the 1st component of the ordered pair.
domain A : 0, 1, 4
Range: The values of the 2nd component of the ordered pair.
range  A :  2, 1, 0, 1, 2
The Rectangular Coordinate System
Ordered Pair
(x, y)
(1st component, 2nd component)
(independent variable, dependent variable)
(input, output)
(abscissa, ordinate)
The Rectangular Coordinate System
y
2nd Quadrant
1st Quadrant
  x, y 
 x, y 
x
  x,  y 
 x,  y 
3rd Quadrant
4th Quadrant
The Rectangular Coordinate System
A  0,0, 1,1, 1,1, 4,2, 4,2
y
x
Graphs, Relations, and Functions
Defn: A function is a relation where every x value has one
and only one value of y assigned to it.
x
y
x
y
x
y
1
3
4
2
2
3
2
5
-3
8
5
7
-4
6
6
1
3
8
1
4
-1
9
-2
-5
3
3
5
6
8
7
domain :  4, 1, 2, 3
range : 3, 4, 5, 6
domain :  3, 1, 4, 5, 6
range : 1, 2, 6, 8, 9
domain :  2, 2, 3, 5, 8
range :  5, 3, 7, 8
Graphs, Relations, and Functions
Functions and Equations.
y  2x  3
y  x2
x  y2
x
y
x
y
x
y
0
-3
2
4
1
1
5
7
-2
4
1
-1
-2
-7
-4
16
4
2
4
5
3
9
4
-2
3
3
-3
9
0
0
function
function
not a function
The Rectangular Coordinate System
y  2x  3
y
x
y
0
-3
5
7
-2
-7
4
5
3
3
x
The Rectangular Coordinate System
y  x2
y
x
y
2
4
-2
4
-4
16
3
9
-3
9
x
The Rectangular Coordinate System
x  y2
y
x
y
1
1
1
-1
4
2
4
-2
0
0
x
The Rectangular Coordinate System
y
x
Graphs, Relations, and Functions
Function Notation
Shorthand for stating that an equation is a function.
Defines the independent variable (usually x) and the
dependent variable (usually y).
y  3x  1
y  x   3x  1
f x  3x  1
y  yx   f x 
Function notation also defines the value of x that is to be use
to calculate the corresponding value of y.
f x   2 x  5
f 3  23  5
f 3  1
3, 1
Graphs, Relations, and Functions
Function Notation – Restricted Domains
4
y  f x  
x
x0
y  f x   2 x  4
2x  4  0
y  f x   4  x
4  x2  0
2