4.
Introduction
to
Inequalities
Understanding
&
Solutions
5.
Inequalities
ï‚› The
Prefix 'in' means not. Incorrect, Inflexible
ï‚› Equations which have solutions are equal to a
specific value, or number: 2x = 8 can only
equal 4; no other number will satisfy this
equation.
ï‚› Inequalities, however, can have many answers.
They are not equal to a specific value.
ï‚› When solving inequalities, we are solving for a
range of numbers, not just one.
ï‚› Let's look at some examples of inequalities
6.
Inequalities
Look at, and think about, the following signs:
The problem is, none of these signs say what
they're really supposed to say. Not only that,
they are all incorrect. To be correct, they needed
to include an inequality.
7.
Inequalities
Let's put this sign in mathematical terms:
Let h = the height required to use the ride. The sign
says you must be 46" tall, therefore h = 46"
According to the sign, if you're not 46" tall, you cannot
ride. But how many people are exactly 46" tall?
What they really meant to say was...
You must be at least 46" tall, or in
mathematical terms...
Your height must be equal to or
greater than 46". This is our
inequality. Our solution is not a
single number, but a range of
numbers.
8.
Inequalities
This sign obviously refers to the drinking age. But
the sign states that even 22 year olds, or 75 year
old people cannot enter. The two words missing
here are: at least
In mathematical terms, the drinking age is:
Equal to or greater than 21
d > 21
9.
Inequalities
As far as the signs are written:
Incorrect
Correct
11.
Solving Inequalities
The process of solving Inequalities is the same as
equations except for one rule(which we'll get to
later), and how inequalities are shown graphically.
Less Than; shown with an open circle on
number line; x < -4
Less Than or equal to; shown with closed
circle on number line; x < -4
12.
Solving Inequalities
Greater Than; shown with an open circle
on number line; x > -4
Greater Than or equal to; shown with a
closed circle on number line; x < -4
13.
Solving Inequalities
Basic Inequalities
1. Write the inequality shown below
x<3
x>0
-5 < x < 2
14.
Inequalities
Graphing Inequalities
Draw a number line and graph the following:
1. 1 <x < 8
2, -2 < x < -1
3. -5 < x
<2
15.
Solving Inequalities
Solve for x and Graph
1. 6x - 7 < 5
1. x < 2; Graph
2. 4(x - 2) > 20
x>3
3. x - 8 < - 6
x<2
And now the one difference between equations &
inequalities:
Solve for x and Graph
4. -2x < 4; When multiplying or dividing by a negative
coefficient, you must switch the sign
4. -2x < 4; -2x/-2 > 4/-2; x > -2
16.
Inequalities
Think about the rule for example 4 with numbers in there,
instead of variables. -2 < 4
You know that the number four is larger than the number
negative two: 4 > -2.
Multiplying through this inequality by â€“1, we get â€“4 < â€“
2, which the number line shows is true:
If we hadn't flipped the inequality, we would have ended
up with "â€“4 > â€“2", which clearly isn't true.
When multiplying or dividing a negative coefficient,
you must flip the sign for the inequality to remain true.
17.
Solving Inequalities
Last 2 Practice Problems;
Solve & Graph on Number Line
5. x - 12 < -6
5. x - 12 < -6;
+12 +12
6. 6 - 2x > - x
+2x +2x
6 > x; x < 6
6. 6 - 2x > - x
5. x < 6;
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