Transcript Chapter 2.7 Inequalitities

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© 2002 by Shawna Haider
There are two kinds of notation
for graphs
Remember---these
mean
theofsame
inequalities: open circle or filled in circle notation
thing---just
twobrackets.
different
notations.
and interval notation
You should
be
familiar with both.
x  1
[
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
circle filled in
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
squared end bracket
Both of these number lines show the inequality
above. They are just using two different notations.
Because the inequality is "greater than or equal to"
the solution can equal the endpoint. That is why the
circle is filled in. With interval notation brackets, a
square bracket means it can equal the endpoint.
Remember---these
Let's look at the two differentmean
notationsthe
with same
a
different inequality
sign.different notations.
thing---just
two
x  1
)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
circle not filled in
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
rounded end bracket
Since this says "less than" we make the arrow go the
other way. Since it doesn't say "or equal to" the
solution cannot equal the endpoint. That is why the
circle is not filled in. With interval notation brackets, a
rounded bracket means it cannot equal the endpoint.
Skill Practice
x > -1
7
c
3
3>y
Compound Inequalities
Let's consider a "double inequality"
(having two inequality signs).
2 x3
(
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
]
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
I think of these as the "inbetweeners".
x is inbetween the two numbers. This is an "and"
inequality which means both parts must be true. It
says that x is greater than –2 and x is less than or
equal to 3.
Skill Practice
1
 x4
2
0 ≤ y ≤ 8.5
Just like graphically there are three different
notations, when you write your answers you can use
inequality notation, set builder notation or interval
notation. Again you should be familiar with both.
[
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
x  1
Inequality notation
for graphs shown
above.
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
{x | x≥ -1}
[1, )
Set builder notation Interval notation
for graphs shown
for graphs shown
above.
above.
Let's have a look at the interval notation.
[1, )
[
unbounded
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
For interval notation you list the smallest x can
be, a comma, and then the largest x can be so
solutions are anything that falls between the
smallest and largest.
The bracket before the –1 is square because this is greater than
"or equal to" (solution can equal the endpoint).
The bracket after the infinity sign is rounded because the
interval goes on forever (unbounded) and since infinity is not
a number, it doesn't equal the endpoint (there is no endpoint).
Let's try another one.
(2,4]
Rounded bracket means
cannot equal -2
Squared bracket means
can equal 4
The brackets used in the interval notation above
are the same ones used when you graph this.
(
]
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
This means everything between –2 and 4 but not including -2
Let's look at another one
(,4)
)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Notice how the bracket notation for graphing
corresponds to the brackets in interval notation.
Remember that square is "or equal to" and round is up
to but not equal. By the infinity sign it will always be
round because it can't equal infinity (that is not a
number).
Addition and Subtraction Properties of Inequality.
Let a, b, and c represent real numbers
1. Addition Property of Inequality:
If a < b,
Then a + c < b + c
2. Subtraction Property of inequality
If a < b
Then a – c < b - c
SOLVING A LINEAR
INEQUALITY
-2p + 5 < -3p + 6
-2p + 5 < -3p + 6
-2p + 5 +3p < -3p +3p + 6
p+5<6
p+5-5<6–5
p<1
Addition property of inequality
(add 3p to both sides).
Simplify
Subtraction property of inequality.
Set-Builder notation: {p | p < 1}
Interval notation: (-∞, 1)
Properties of Inequalities.
Essentially, all of the properties that you learned to
solve linear equations apply to solving linear
inequalities with the exception that if you multiply or
divide by a negative you must reverse the
inequality sign.
So to solve an inequality just do the same steps as
with an equality to get the variable alone but if in the
process you multiply or divide by a negative let it ring
an alarm in your brain that says "Oh yeah, I have to
turn the sign the other way to keep it true".
Example:
2x  6  4x  8
- 4x
- 4x
 2x  6  8
+ 6 +6
 2 x  14
-2
We turned the sign!
-2
x  7
Ring the alarm!
We divided by a
negative!
Skill Practice Solving a Linear
Inequality
-5x -3 ≤ 12
Add 3 to both sides
-5x -3 +3 ≤ 12 +3 Divide by -5. Reverse the
direction of the inequality sign.
-5x ≤ 15
-5x ≥ 15
-5
-5
x ≥ -3 or [-3, ∞)
Solving a Compound
Inequality of the Form
a<x<b
-3 ≤ 2x + 1 < 7
-3 -1 ≤ 2x +1 -1 < 7 -1
-4 ≤ 2x < 6
-4 ≤ 2x < 6
2
2 2
Subtract -1 from all three
parts of the inequality
Divide by 2 in all three
parts of the inequality
-2 ≤ x < 3 or [ -2, 3)
Applications of Linear
Inequalities
English Phrase
Mathematical Inequality
a is less than b
a<b
a is greater than b
a exceeds b
a is less than or equal to b
a is at most b
a is no more than b
a is greater than or equal to b
a is at least b
a is no less than b
a>b
a≤b
a≥b
Translating Expressions
Involving Inequalities
Bill needs a score of at least 92 on the final
exam. Let x represent Bill’s score
x ≥ 92
Fewer than 19 cars are in the parking lot. Let
c represent the number of cars. c < 19
The heights, h, of women who wear
petite sixe clothing are typically
58 ≤ h ≤ 63
between 58 in. and 63 in, inclusive.
Solving an Application with
Linear Inequalities
To earn an A in a math class, Alsha must
average at least 90 on all of her tests.
Suppose Alsha has scored 79, 86, 93, 90,
and 95 on her first five math tests.
Determine the minimum score she needs
on her sixth test to get an A in the class
x ≥ 97