You already know inequalities. Often they are used to place limits on variables. That just means x...
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Transcript of You already know inequalities. Often they are used to place limits on variables. That just means x...
x 12
9x
x 12
9x
x 12
9x
x 129x
912 x
9x
You already know inequalities.
Often they are used to place limits on variables.
That just means x can be any number equal to 9 or less than
9.
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.
7 and 9 xx
7 and 9 xx
Let’s look at the graph:
0 5 10 15-20 -15 -10 -5-25 20 25
The upper limit is 9. Because x can be equal to 9, we mark it with a filled-in
circle.
0 5 10 15-20 -15 -10 -5-25 20 25
The lower limit is -7. We also need to mark it with a filled-in
circle.
7 and 9 xx
Where are they found on the graph?
Yes!
It is less than or equal to 9?What about –15?
0 5 10 15-20 -15 -10 -5-25 20 25
There are other numbers that satisfy both conditions.
7 and 9 xx
15
It is also greater than or equal to -7?
What about –15?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
15 No!
Where are they found on the graph?
Because the word and is used, a number on the graph
needs to satisfy both parts of the inequality.
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
15
Yes!720
So let’s try 20. Does 20 satisfy both conditions?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
20
?
15
No!920
So let’s try 20. Does 20 satisfy both conditions?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
20
?
15
Since 20 does not satisfy both conditions, it can’t
belong to the solution set.
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015
There is one region we have not checked.
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7 and 9 xx
2015
We need to choose a number from that region.
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015
You want to choose 0?Good choice! 0 is usually the easiest number to work
with.0
Does 0 satisfy both conditions?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015 0
70 Yes!?
Does 0 satisfy both conditions?
0 5 10 15-20 -15 -10 -5-25 20 25
7 and 9 xx
2015 0
90 Yes!?
If one number in a region completely satisfies an
inequality,
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7 and 9 xx
2015 0
you can know that every number in that region
satisfies the inequality.
Let’s graph another inequality:
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1or 12 xx
tells us we want an open circle,The first sign
First we mark the boundary points:
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
and the 12 tells us where the circle goes.
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
and the 12 tells us where the circle goes.
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
tells us we want a closed circle,The second sign
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
and the -1 tells us where the circle goes.
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
The boundary points divide the line into three regions:
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
1 2 3
We need to test one point from each region.
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1or 12 xx
10 1210
No!? 1210
Yes!?
Notice that the word used is or,
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1or 12 xx
10 1210
No! 1210
Yes!
instead of and.
Or means that a number
0 5 10 15-20 -15 -10 -5-25 20 25
1or 12 xx
10 1210
No! 1210
Yes!
only needs to meet one condition.
Because –10 meets one condition,
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1or 12 xx
1210 1210 Yes!
10
the region to which it belongs . . .. . . belongs to the graph.
Let’s check the next region:
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1or 12 xx
125 15 5
No!? No!?
Because –1 meets neither condition,
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1or 12 xx
125 15 5
No! No!
the numbers in that regionwill not satisfy the inequality.
Now the final region:
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1or 12 xx
1215 115 15
Yes!? No!?
Again, 15 meets one condition
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1or 12 xx
1215 115 15
Yes!
so we need to shade that region.
A quick review:
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1. Find and mark the boundary points.2. Test points from each region.3. Shade the regions that satisfy the inequality.? ? ?
To graph a compound inequality:
A quick review:
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1. Find and mark the boundary points.2. Test points from each region.3. Shade the regions that satisfy the inequality. or
Given the graph below, write the inequality.
First, write the boundary points.
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15 20
15
Then look at the marks on the graph,
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20 15 20
and write the correct symbol.
Since x is between the boundary points on the
graph,
15 20
0 5 10 15-20 -15 -10 -5-25 20 25
it will be between the boundary points in the inequality.
xx
20 15 x
Since x is between the boundary points on the
graph,
x0 5 10 15-20 -15 -10 -5-25 20 25
it will be between the boundary points in the inequality.
0 5 10 15-20 -15 -10 -5-25 20 25
Try this one:Again, begin by writing the
boundary points:
10 5
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And again, you need to choose the correct symbols:
10 5
0 5 10 15-20 -15 -10 -5-25 20 25
Because the x-values are not between the boundary points
on the graph,we won’t write x between the
boundary points in the equation.
x x 5 10
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x xx105x
Because the x-values are not between the boundary points
on the graph,we won’t write them between
the boundary points in the equation.
0 5 10 15-20 -15 -10 -5-25 20 25
x xx105x
We will use the word, or, instead:Remember that or means
a number has to satisfy only one of the conditions.
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x x
We will use the word, or, instead:Remember that or means
a number has to satisfy only one of the conditions.
xx 10or 5
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Is there any one number that belongs to both shaded
sections in the graph?
xx 10or 5
Say NO!NO!
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xx 10or 5
So it would be incorrect to use and. And implies that a
number meets both conditions.
. . . you remember that a compound inequality is just two
inequalities put together.
9325 x
Solving compound inequalities is easy if . . .
932 x325 x 9325 x
9325 x
You can solve them both at the same time:
3 3628 x
2 2
3
234 x
510 x
Write the inequality from the graph:
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1: Write boundaries:2: Write signs:3: Write variable:
Solve the inequality:
57415 x
Is this what you did?
771248 x
4
7
4 42 x 3
You did remember to reverse the signs . . .
57415 x771248 x
4
7
4 42 x 3
. . . didn’t you?Good job!