Transcript Inequalities.PPT

Solving Inequalities
Objective- To solve and graph simple
inequalities involving
addition/subtraction.
Equations
Inequalities
Solve and graph.
x2 5
2  2
x 3
-3 -2 -1 0 1 2 3 4
One Solution
Solve and graph.
x2 5
2  2
x 3
-3 -2 -1 0 1 2 3 4
Infinite Solutions
Graph the following inequalities.
1) x  2
-3 -2 -1 0 1 2 3 4
2) x  3
-3 -2 -1 0 1 2 3 4
3) x  2
-3 -2 -1 0 1 2 3 4
4) x  1
-3 -2 -1 0 1 2 3 4
Solve and graph the inequalities.
1) 3  x  7
2) 4  x  6
3
3
6
6
x  10
2x
x 2
-10
0
10
-1
0
1
2
3
Solve and graph the inequalities.
3) 5  8  k
4) 3  p  9
8  8
3
3
13  k
p 12
k  13
-13
0
0
12
Objective- To solve and graph simple
inequalities involving
multiplication/division.
Solve and graph the inequalities.
x
1) 4x  20
2)
 2
3
4x  20
x
(3)  2(3)
4 4
3
x 5
x  6
0
5
-6
0
6
Inequalities transform like equations except...
3x  6
3x  6
3  3
x  2
reverse
x  2
Inequalities transform like equations except...
When multiplying or dividing by a negative
number you must reverse (flip) the inequality.
-4 -3 -2 -1
0
1
Negative side
3  2
Large is small
2
3
4
Positive side
3 2
reverse
(1)3  2(1)
3  2
3 2
Large is large
Solve and graph the inequalities.
y
1) 4x  24
2)
7
3
4x  24
y
(3)
 7(3)
4  4
3
x  6
y  21
-6
0
-21
0
Solve and graph the inequalities.
m
3) 3x  18
4)   6
2
3x  18
m
(2)
 6(2)
3
3
2
x  6
m  12
-6
0
-12
0
Objective- To solve and graph multistep
inequalities.
Solve and graph.
2x  5 11
5  5
2x 16
2 2
x 8
0
8
Solve and graph.
1) 3x  4 17
4  4
3x  21
3 3
x 7
0
7
x
2)
5 3
2
5  5
x
8
2
x
(2)  8(2)
2
x  16
0
16
Solve and graph.
3) 7  x  4
7
7
x  3
(1)( x)  (3)(1)
x 3
0
4) 3(x  5)  12
3x 15 12
15 15
3x  27
3 3
x 9
3
0
9
Solve and graph.
5) 3(x  4)  9
3x 12  9
12 12
3x  21
3x  21
3
3
x 7
0
7
6) 4x  3  2x 11
2x
 2x
6x  3 11
3  3
6x 14
6 6
7
x
3
1
2
0
3
Joe is saving for a new $500 bike. He currently
has $125. If he saves $15 per week, how long
must he wait to save at least $600 to cover tax
and extras.
Let x = # of weeks
125 15x  600
125
125
15x  475
15
15
2
x  31
3
At least 32 weeks
A water tank contains 500 gallons of water and
begins leaking at a rate of 4 gallons per minute.
Because the tank must be repaired from the
inside, it can retain at most 60 gallons of water.
How long must the repairman wait to fix the tank?
Let x = # of minutes tank leaks
500  4x  60
500
 500
4x  440
4x  440
4
4
x  110
At least 110 min.
Objective- To solve compound
inequalities involving “or”.
Write a compound inequality that describes
all real numbers less than -2 or greater than 5.
x  2
x 5
or
Graph.
-3
-2
-1
0
1
2
3 4
5
6
Graphing Unions (OR)
1. Graph both inequalities on the same line.
2. If graphs overlap, then solution is all real
numbers (whole # line).
R  all real numbers
Solve and graph the compound inequality.
2x  3  7
3  3
2x 10
2
2
x 5
or
or
4x  7  33
7  7
4x  40
4
4
x  10
x  5 or x  10
0
5
10
Solve and graph the compound inequality.
or 1 2x  23
5x  35
5x  35
5  5
1
x  7
x  7 or x  12
-7
0
1
2x  24
2x   24
2
2
x  12
12
Solve and graph.
4( x  5)  4 or 2( x  4)  16
2x  8  16
4x  20  4
20 20
8 8
4x  16
2x  24
4
4
2
2
x

12
x4
x  4 or x  12
0
4
12
Solve and graph.
3x  5  7 or 8x 14  66
5 5
14 14
3x  12
8x  80
3
3
8
8
x4
x  10
x  4 or x  10
0
4
10
x = All Real #’s (R)
Objective- To solve compound
inequalities involving “and”.
Write a compound inequality that describes all
the real numbers greater than -2 and less than 5.
x  2
-3
-2
-1
x 5
and
0
1
2
3 4
5
6
Graphing Intersections (AND)
1. Graph the two inequalities separately (lightly).
2. Find where the graphs overlap.
3. Graph only the overlapping part.
4. If there’s no overlap, then there’s no solution.
 = No Solution
Solve and graph.
3x  5  7 and 8x 14  66
5 5
14 14
3x  12
8x  80
3
3
8
8
x4
x  10
x  4 and x  10
0
4
10
Solve and graph.
2x  1  3 and 3x  4  17
1 1
4 4
2x  2
3x  21
2 2
3
3
x 1
x7
x  1 and x  7
0 1
7
Another Way to Write “AND”
“In-between
statement”
2  x  5
x  2
-3
-2
-1
x 5
and
0
1
2
3 4
5
6
Solve and graph the compound inequality.
4  x  3  7
4  x  3  7
4  x  3 and x  3  7
3
3
3  3
x 4
7  x
x  7 and x  4
4  x  3  7
3  3  3
7  x  4
7  x  4
-7
0
4
Solve and graph.
5  3x  4  7
4  4  4
9  3x  3
3 3 3
3  x  1
-3
0
1
Solve and graph.
4  6  x  8
6  6  6
10  x  2
10  x  2
1 1 1
10  x  2
2  x 10
-2
0
10
Solve and graph.
7  2x  5  9
5  5
5
12  2x  4
2
2
2
6  x  2
-6
0
2
A racquetball club charges a $20 membership
and $2 per hour. How many hours per month
can be played on a budget of $50 to $70?
Let h = # of hours
Cost = 20 + 2h
50  Cost  70
50  20  2h  70
20  20
 20
30  2h  50
2
2
2
15  h  25
From 15 to 25 hours
Number Line Graphs of Inequalities
Intersections
Unions
x < 5 and x < 3
x < 5 or x < 3
0 1 2 3 4 5 6
x<3
3<x<5
5 6
x<5
x < 5 and x > 3
0 1 2 3 4
0 1 2 3 4
5 6
x < 5 or x > 3
0 1 2 3 4
5 6
x = Any Real Number
Number Line Graphs of Inequalities
Intersections
Unions
x > 5 and x < 3
x > 5 or x < 3
0 1 2 3 4 5 6
Not possible
x > 5 and x > 3
0 1 2 3 4
x>5
5 6
0 1 2 3 4
5 6
x < 3 or x > 5
x > 5 or x > 3
0 1 2 3 4
x>3
5 6
Number Line Graphs of Inequalities
Intersections
Unions
x < 4 and x < 7
x < 4 or x < 7
0 2 4 6 8 10 12
x<4
x < 4 and x > 7
0 2 4 6 8 10 12
Not possible
0 2 4 6 8 10 12
x<7
x > 4 or x < 7
0 1 2 3 4
5 6
x = Any Real Number
Equalities
= Equals- is the
same as
 Congruent- same
size and shape
~ Similar- same
shape
Inequalities
< Is less than
> Is greater than
 Is less than or
equal to
 Greater than or
equal to.
 Approx. equal to
= Not equal to