Transcript Inequalities

Math 374
Inequalities
Topics Covered
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1) Number Lines
2) Inequality Sign
3) Inequality Form
4) Interval Form
Number Lines
• People have always turned to pictures
to help them visualize concepts.
• The concept of numbers has
traditionally been represented by a
line.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Lines Notes
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
• We consider zero to be the centre with no
sign
• We say positive extends to the right and
negative extends to the left
Number Line Notes
• The negatives are a mirror image of the
positives and vice versa
• We say the numbers continue to the right
to positive infinity represented by + ∞
• Numbers continue to the left to negative
infinity represented by - ∞
• We can represent all the numbers that we
know this way
Notes
• Consider ¾
0
¾
1
Enclosing Integers
How many numbers are between 0 & 1?
The answer is ∞
Notes
• What is the next number after 1?
• The answer is impossible to determine.
• Consider…
5 6 7
• We say 5 is less than 7 because 5 falls to
the left of 7. We need a symbol.
Notes
• < or >
• The symbol has two sides
• Remember that the direction that the sign
“hugs” is bigger.
• Thus 5 < 7 (5 is less than 7)
• 7 > 5 (7 is larger than 5)
• The two statements mean the same but
use different symbols
Notes
• To make this work, we will always
work by reading left to right
• We want to handle this like equations
• Let us look at what we do
• 5<7
• (Adding 5) 5 + 5 < 7 + 5
• 10 < 12 (still true)
Notes
• (Subtract 5) 5 – 5 < 7 – 5
• 0 < 2 … still true
• This operation is called transposing
and so this works!
• (Multiply 5) 5 x 5 < 7 x 5
• 25 < 35 still true … what is next?
Notes
• (multiply by -5) 5 x (-5) < 7 x (-5)
• -25 < -35 is FALSE
• To make it true, we must reverse the
inequality sign.
• Therefore -25 > -35 now true
Notes
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(Divide by 2) 5 / 2 < 7 / 2
2.5 < 3.5 still true
(divide by -2) 5 / -2 < 7 / -2
-2.5 < - 3.5 FALSE
To make it true we must reverse the
inequality sign. Thus – 2.5 > - 3.5
Final Important Notes
• 1) We will always move our x to the left
side
• 2) Whenever we multiply or divide by a
negative number, we reverse the inequality
sign
• Additional symbols
• ≤ Greater than or equal to
• ≥ Less than or equal to
Filled means included
Unfilled means not included
Exercises
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5x – 4 ≥ 2x + 11
3x ≥ 15
x ≥ 5 (This is inequality form)
4x – 7 < 8x + 9
-4x < 16
x > -4
Other Forms
• Graph Form
• We reproduce a miniature number
line
• Ex x ≥ 5
-∞
5
+∞
Notes
• Ex x < - 3
-3
• Ex. x > ¾
Note ¾ = 0.75
• Ex x ≤ -42
3
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Note that this is -21.5
Notes
• Intervals and Gaps
• We can work on the number line and
create some strange situations. The
key are the end points
Interval
Form
3
8
Notes
• This is a GRAPH FORM INTERVAL
• We have all the numbers between 3
and 8 including 8 but not including 3.
• Using inequality form…
• 3<x≤8
• Using Interval Form
3
8
Notes
• Interval Form
• ]3, 8]
• Think hugging and not hugging. If it
does not hug the 3, it does not
include the 3.
• If it hugs the 8, it includes the 8.
• You can never equal ∞
Notes
• Consider
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-∞
∞ Graph Form
6
• Inequality Form… x ≥ 6
• Interval Form [6, ∞ [
• Do #1 a – y (a, v & y as examples)
Notes
• Given one form, give other forms…
• Consider ] - ∞, - 7 ] Interval form
• Graph form
-7
• Inequality form x < - 7
New Symbol!
• Consider x ≤ - 5 U x > 7 (U means union)
• State interval form
-5
7
Notes
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State the Interval form
- ∞, 5 ] U ] 7, ∞
Consider 6x – 5 ≥ 9x + 5
-3x ≥ 10
X ≤ -10
3
] -∞, - 10
3
Notes
• In graph form…
-4
-10
3
• Do #2 a – j ; #3 a j
-3
Standard Form
• Consider 5y – 3x ≥ 2
• 5y ≥ 3x + 2
• y ≥ 3x + 2
5
5
Standard Form
• 9x – 7y – 100 < 0
• - 7y < -9x + 100
• y > - 9x + 100
-7
-7
y > 9x – 100
7
7
Notice the inequality changed signs AND notice
what happened to the sign because a + divided by
a – becomes -.
Do #4 a - j