Transcript Inequalities & Interval Notation

Inequalities &
Interval Notation
ES: Demonstrate understanding of
concepts
Objective
To examine the properties of inequalities.
 To express inequalities in interval notation.

Vocabulary

Real Numbers: The set of numbers consisting of
the positive numbers, the negative numbers,
and zero.

Rational Number: A real number that can be
expressed as a ratio of two integers.

Irrational Number: A real number that can not
be expressed as a ratio of two integers.
Rational or Irrational ???
3
4
Rational
Any rational number can be written as a fraction.
Rational or Irrational ???
365
365  
1
Rational
Any integer can be written as a fraction.
Rational or Irrational ???
5
0.625 
8
Rational
Any terminating decimal can be written as a fraction.
Rational or Irrational ???
2
0.222 
9
Rational
Any repeating decimal can be written as a fraction.
Rational or Irrational ???
1.732050808...
Irrational
Irrational numbers can be represented by decimal numbers
in which the digits go on forever without ever repeating.
Rational or Irrational ???
5
Irrational
Some of the most common irrational numbers are radicals.
Rational or Irrational ???
3
125  5
Rational
Be careful, not all radicals are irrational.
Rational or Irrational ???
2
3
Irrational
Numbers containing  are always irrational.
Rational or Irrational ???
1
0.142857 
7
Rational
Remember, any repeating decimal can be written as a fraction.
Rational or Irrational ???
8e
Irrational
Numbers containing the mathematical constant e
(Euler’s number  2.718) are always irrational.
Vocabulary

Real Number Line: A line that pictures
real numbers as points.

All real numbers (rational/irrational) can
be graphed on a number line.
 32
2.6
5
4
3
2
origin
1
0
e 
2
1
2
3
4.3
4
5
Inequalities
Math Wild Kingdom
The greedy crocodile
always wants to eat
the larger thing.
Inequalities
A  B
Less than
Greater than
(smaller)
(larger)
A  B
Greater than
Less than
(larger)
(smaller)
The arrow > points from the greater value to the lesser.
Inequalities
Transitive Property
A  B
B  C

A  C
Inequalities
A  B
AC
 BC
What happens to the inequality sign when you add or subtract?
The inequality remains the same.
AC
 B C
Inequalities
3 
2
What happens to the inequality sign when you multiply by 5?
 3 5

 2  5
15

10
Inequality sign
is still correct
Inequalities
3 
2
What happens to the inequality sign when you multiply by -5?
 3 5

 2  5
15

10
Inequality sign is
no longer correct
Inequalities
3 
2
What happens to the inequality sign when you multiply by -5?
 3 5

 2  5
15

10
Inequality sign
must get flipped
Inequalities
Classic Mistake
Inequalities
What does this mean? What x-values is it talking about?
3  x  2
x exists between -3 and 2
x is less than and
can equal 2
x is larger than, but
cannot equal -3
< excludes the endpoint
< includes the endpoint
Inequalities
3  x  2
(
3
x exists between -3 and 2
]
2
Parentheses: endpoint is not allowed as a value
Bracket: endpoint is allowed as a value
Interval Notation
3  x  2
x exists between -3 and 2
]
(
3
Same as
2
( 3 , 2 ]
Interval excludes -3, and includes 2
Interval Notation
1  x  10
[
]
1
10
[ 1 , 10 ]
Interval Notation
3  x  0
)
)
3
0
( 3 , 0 )
Interval Notation
x5
[
5
[ 5 , )
Always use parentheses with .
Interval Notation
x  1
)
1
( , 1)
Always use parentheses with .
Interval Notation
What values below does this expression represent?
 ,  
0
(-1, 1)
Nothing
All values
Interval Notation
What values does this expression represent?
 ,  
This represents all values on the line.
Interval Notation
What does this mean? Is there anything wrong with the notation?
 , 4 
Never use a bracket with 
Interval Notation
 5, 2 
What would the inequality notation look like?
5  x  2
Interval Notation
 0, 1
What would the inequality notation look like?
0  x 1
Conclusion

When increasing/decreasing two sides of an inequality
by the same amount, the inequality remains.

When multiplying/dividing an inequality by a negative,
the inequality sign flips.

Use a bracket if the inequality symbol next to the
number is < or >, otherwise use a parenthesis.

Always use parentheses with  and - .
Exit Slip: Answer the below questions on the note card
then turn in. Make sure your name is on it.
1)
Circle all that apply:
a) -5 is… Real Rational
b)
c)
3 is…
is…
81
Real
Real
Rational
Rational
Irrational
Irrational
Irrational
2) Write the interval notation for each of the below
a) - 4 < x
b) 2 < x ≤ 5
c) x ≥ 0
3) Write the interval notation for the graph below which represents all
real numbers
0
4) Solve the inequality and write the solution in interval notation
-3x + 2 < 11