Solving Inequalities
Solving Inequalities
Objective- To solve and graph simple inequalities involving addition/subtraction.
Equations InequalitiesSolve and graph. Solve and graph.
x 2 5
2 2
x 3
-3 -2 -1 0 1 2 3 4
One Solution
x 2 5
2 2
x 3
-3 -2 -1 0 1 2 3 4
Infinite Solutions
Graph the following inequalities.
1)
2)
3)
4)
x 2
x 3
x 2
x 1
-3 -2 -1 0 1 2 3 4
-3 -2 -1 0 1 2 3 4
-3 -2 -1 0 1 2 3 4
-3 -2 -1 0 1 2 3 4
Solve and graph the inequalities.
1) 2)
3 x 7
4 x 6
3 3
x 10
-10 0 10-1 0 1 2 3
6 6
2 x
x 2
Solve and graph the inequalities.
3) 4)
5 8 k
3 p 9
8 8
13 k
-13 0 0 12
3 3
p 12
k 13
Objective- To solve and graph simple inequalities involving
multiplication/division.
4x 20
Solve and graph the inequalities.
1) 2)
4x 20
x3
2
x 5
0 5
-6 0 6
(3)x3
2(3)
x 6
4 4
Inequalities transform like equations except...
3x 6
3 3
3x 6
reverse
x 2
x 2
( 1)3 2( 1) reverse
Inequalities transform like equations except...
When multiplying or dividing by a negativenumber you must reverse (flip) the inequality.
-4 -3 -2 -1 0 1 2 3 4
Positive sideNegative side
Large is largeLarge is small3 2 3 2
3 2
3 2
4x 24
Solve and graph the inequalities.
1) 2)
4x 24
y 3
7
x 6
-6 0
-21 0
( 3)y 3
7( 3)
y 21
4 4
Solve and graph the inequalities.
3) 4)
3x 18
m2
6
x 6
-6 0
-12 0
( 2)m 2
6( 2)
m 12
3 3
3x 18
Objective- To solve and graph multi-step
inequalities.
Solve and graph.
2x 5 11
5 5
2x 16
2 2
x 80 8
1) 2)
3x 4 17
4 4
3x 21
3 3
x 7
0 7
x2
5 3
5 5
x2
8
(2)x2
8(2)
x 160 16
Solve and graph.
3) 4)
7 x 4
7 7
x 3
0 3
3(x 5) 12
3x 15 12
3x 27
x 9
0 9
Solve and graph.
( 1)( x) ( 3)( 1)
x 3
15 15
3 3
5) 6)
3(x 4) 9
12 12
3x 21
0 7
4x 3 2x 11
6x 3 11
x 73
Solve and graph.
3x 3
21 3
x 7
2x 2x
6 6
3x 12 9
3 3
6x 14
0
213
Joe is saving for a new $500 bike. He currentlyhas $125. If he saves $15 per week, how long must he wait to save at least $600 to cover taxand extras.
Let x = # of weeks
125 15x 600
125 125
15x 475
15 15
x 3123
At least 32 weeks
A water tank contains 500 gallons of water andbegins 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 4
440 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 or
x 5
-3 -2 -1 0 1 2 3 4 5 6
Graph.
Graphing Unions (OR)
1. Graph both inequalities on the same line.
2. If graphs overlap, then solution is all real numbers (whole # line).
all real numbersR
2 2
4 4
Solve and graph the compound inequality.
2x 3 7
4x 7 33or
3 3
2x 10
x 5 or
7 7
4x 40
x 10
x 5 or
x 10
0 5 10
2x 24
5x 35
5 5
2 2
Solve and graph the compound inequality.
5x 35
1 2x 23or
x 7
or
1 1
2x 24
x 12
-7 0 12
x 7
x 12
Solve and graph.
4( 5) 4x or 2( 4) 16x
20 204 16x 4 4
4x
8 82 24x2 2
12x
4 12 x or x
0 4 12
4 20 4x 2 8 16x
Solve and graph.
3 5 7x or 8 14 66x 5 5
3 12x 3 3
4x
14 148 80x8 8
10x4 10 x or x
0 4 10
x = All Real #’s (R)
Objective- To solve compound inequalities involving “and”.
Write a compound inequality that describes allthe real numbers greater than -2 and less than 5.
x 2 and
x 5
-3 -2 -1 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.
3 5 7x and 8 14 66x 5 5
3 12x 3 3
4x
14 148 80x8 8
10x
4 10 x and x
0 4 10
Solve and graph.
2 1 3x and 3 4 17x 1 1
2 2x2 2
1x
4 43 21x3 3
7x
1 7 x and x
0 1 7
Another Way to Write “AND”
x 2 and
x 5
-3 -2 -1 0 1 2 3 4 5 6
2 x 5 “In-between statement”
Solve and graph the compound inequality.
4 x 3 7
4 x 3 7
4 x 3 and
x 3 7
3 3
7 x
3 3
x 4
x 7 and
x 4
7 x 4-7 0 4
4 x 3 7
3 3 3
7 x 4
9 3x 3
3 3 3
Solve and graph.
5 3x 4 7
4 4 4
3 x 1
-3 0 1
10 x 2
1 1 1
Solve and graph.
4 6 x 8
6 6 6
-2 0 10
10 x 2
10 x 2
2 x 10
12 2 4x 2 2 2
Solve and graph.7 2 5 9x 5 5 5
6 2x
-6 0 2
A racquetball club charges a $20 membershipand $2 per hour. How many hours per monthcan be played on a budget of $50 to $70?
Let h = # of hours
Cost = 20 + 2h
15 25h
From 15 to 25 hours
50 20 2 70h
30 2 50h 2 2 2
20 20 20
50 Cost 70
Number Line Graphs of InequalitiesIntersections Unions
x < 5 x < 3and x < 5 x < 3or 0 1 2 3 4 5 6 0 1 2 3 4 5 6
x < 3 x < 5
x < 5 x > 3and 0 1 2 3 4 5 6
3 < x < 5
x < 5 x > 3or 0 1 2 3 4 5 6
x = Any Real Number
x > 5 x < 3and x > 5 x < 3or 0 1 2 3 4 5 6 0 1 2 3 4 5 6
Not possible x < 3 or x > 5
x > 5 x > 3and 0 1 2 3 4 5 6
x > 5
x > 5 x > 3or 0 1 2 3 4 5 6
x > 3
Intersections UnionsNumber Line Graphs of Inequalities
Number Line Graphs of InequalitiesIntersections Unions
x < 4 x < 7and x < 4 x < 7or 0 2 4 6 8 10 12 0 2 4 6 8 10 12
x < 4 x < 7
x < 4 x > 7and 0 2 4 6 8 10 12
Not possible
x > 4 x < 7or 0 1 2 3 4 5 6
x = Any Real Number
Equalities Inequalities
= Equals- is the same as
Congruent- same size and shape
Similar- same shape
< Is less than
> Is greater than
Is less than or equal to
Approx. equal to
= Not equal to
~
Greater than or equal to.