Transcript PPT 2.6 Compound Inequalities

Compound Inequalities
Section 2.6
Advanced Algebra 1
Warm-UP Solve and Graph:
1.)
2x - 4 ³ -20
+4
+4
2x ≥ -16
2
2
x ≥ -8
3.)
2.)
-3x < 9
-3
x > -3
4x - 5 > -3x +16
+3x
+3x
7x – 5 > 16
+5
+5
7x ≥ 21
7
7
-3
x>3
What integers are less than 5 and greater than 1?
2, 3, 4
What integers are greater than -2 and less than or equal to 3?
-1, 0, 1, 2, 3
What integers are greater than -3 and less than 4?
-2, -1, 0, 1, 2, 3
What integers are greater than 2 and less than 2?
None!
Sometimes we put more than one
limit on the variable:
Now x is still less than or equal to 9,
but it must also
be greater than or equal to –7.
x  9 and x  7
If one number in a region
completely satisfies an inequality,
you can know that every number
in that region satisfies the
inequality.
 15
-25 -20 -15 -10 -5
20
0
0
5
10 15 20 25
x  9 and x  7
“AND” problems first!
Dealing with more than one inequality at a time.
Find “x” that would satisfy both inequalities.
Graph x ≥ 0
Graph x < 4
What if I wanted to
graph
“x ≥ 0 AND x < 4”
on the same
number line
We can actually re-write this as: 0 ≤ x < 4
“x is between 0 and 4”
We are looking for the overlapping section of both
graphs up top
GRAPH EACH INEQUALITY:
1.) -2 < y < 0
2.) 7 ≤ t ≤ 8
3.) 4 ≤ n ≤ 11
4.) x < 2 AND x < -3
Remember that “AND” means it has to satisfy both inequalities
5.)
-2 < x + 2 ≤ 4
-2
-2 -2
-4 < x ≤ 2
7.) -5 ≤ 2x + 3 < 7
-3
-3 -3
6.) -6 ≤ 3x ≤ 12
3
3
3
-2 ≤ x ≤ 4
8.) -3 < -1 – 2x ≤ 5
-8 ≤ 2x < 4
2 2
2
+1 +1
+1
-2 < -2x ≤ 6
-2 - 2 -2
-4 ≤ x < 2
1 > x ≥ -3
-3 ≤ x < 1
One more:
9.)
1 ≤ -x – 6 ≤ 1
+6
+6 +6
7 ≤ -x ≤ 7
-1 - 1 -1
-7 ≥ x ≥ -7
-7 ≤ x ≤ -7
ON YOUR OWN:
1.) Write an compound inequality that represents the set of
all real numbers greater or equal to -2 and less than 3. Then
graph inequality.
-2 ≤ x < 3
2.) Solve: -5 ≤ 2x + 3 < 7
-3
-3
-8 ≤ 2x < 4
2 2 2
-3
-4 ≤ x < 2
Now “OR” problems
Dealing with more than one inequality at a time.
Find “x” that would satisfy either inequality
Warm-UP: Solve and Graph
1.) -7 < x – 4 < 3
+4
+4
-3 < x < 7
+4
2.) -3 ≤ 2x – 5 < 7
+5
+5
+5
2 ≤ 2x < 12
2 2
2
1≤x<6
What integers are less than 5 or greater than 1?
All integers
What integers are greater than -2 or greater than 3?
All integers greater than -2
What integers are less than -3 or greater than 4?
x < -3 or x > 4
What integers are greater than 2 or less than 2?
All integers except 2
Graph x > 2
Graph x < -1
What if I wanted to
graph
“x > 2 OR x < -1”
on the same number
line
We are looking for the UNION of all sections shaded
Or means that a number
only needs to meet one condition.
Can you think of any numbers that satisfy one of these
conditions?
-25
-20
-15
-10
-5
0
5
10
15
20
x  12 or x  1
25
Graph each compound inequality:
1.) x < 0 or x > 5
3.) x – 4 ≤ 3 or 2x > 18
+4 +4
2
2
x≤7
OR x > 9
2.) x ≤ -1 or x ≥ 7
4.) 3x + 1 < 4 or 2x – 5 > 7
+5 +5
-1 -1
2x > 12
3x < 3
3
x<1
3
2
2
x>6
Look at the graph on the number line and create
the inequality.
First look at the boundary points.
-25
-20
-15
-10
-5
0
5
10
15
 15  x  20
20
25
Try this one:
Again, begin by writing the
boundary points:
-25
-20
-15
-10
-5
0
5
10
15
20
x  5 or 10  x
25
Is there any one number that
belongs to both shaded
sections in the graph?
NO!
Say
NO!
x  5 or 10  x
-25
-20
-15
-10
-5
0
5
10
15
20
25
Write the inequality from the
graph:
-25
-20
-15
-10
-5
0
5
10
15
 10  x  5
20
25
Solve the inequality:
 15  4 x  7  5
7
7 7
 8  4 x  12
4 4 4
2  x 3
Final Answer is -3  x < 2
Solve and Graph these
inequalities.
1.) -7  3-x < 5
2.) -25 < -5x < 0
3.) -3  -5-2x < 1
4.) 3  2x+3  7
5.) -6  -3x  12
6.) x-43 or 2x>18
7.) x-4<-8 or x+3>5
8.) 2x+31 or 3x-5>1
ANSWERS
1.)
-2 < x  10
2.) 0 < x < 5
3.) -3 < x  -1
4.) 0  x  2
5.) -4  x  2
6.) x  7 or x > 9
7.) x < -4 or x > 2
8.) x  -1 or x > 2