We are asked to compare the numbers
using the less than or greater
than inequality symbols.
We will make the comparison two ways.
For the first method,
we will plot the numbers
on the number line
and determine which value is on the left
and which value is on the right.
The value on the left
is always less than the value on the right
and the value on the right
is always greater than
the value on the left.
For the second method, we will
obtain a common denominator.
For part A,
we're asked to compare
negative 2/3 and negative one.
Negative 2/3 is 2/3 of a
unit to the left of zero
and falls between zero and negative one.
To plot this fraction accurately,
you will cut the length
from zero to negative one
into three equal parts
where each part has a
length of 1/3 of a unit.
Starting at zero, we have negative 1/3
and then negative 2/3.
And of course negative one is
one unit to the left of zero,
which is located here.
Because negative 2/3 is first
and negative 2/3 is to
the right of negative one,
we say that negative 2/3 is
greater than negative one.
And now to make the comparison
by obtaining a common denominator.
We have negative 2/3
compared to negative one.
Let's write negative one as a fraction
with a denominator of one.
And now we will obtain
a common denominator,
hopefully the least common denominator,
which in this case is three.
We need to write negative one over one
as an equivalent fraction
with a denominator of three.
Therefore, we multiply the numerator
and denominator by three.
This gives us negative 2/3
compared to negative 3/3.
Once we have a common denominator,
we can compare the fractions
by simply comparing the numerators,
as long as we place the
negative sign in the numerator.
Which means to compare
these two fractions,
we only have to compare
negative two and negative three.
Because negative two is
greater than negative three,
this tells us negative 2/3
is greater than negative one.
For the second example,
you want to compare negative
two and negative 1 3/4.
Negative two is two units to
the left of zero, located here.
Negative 1 3/4 is located
one and 3/4 units to the left of zero,
which is located between
negative one and negative two.
To plot this mixed number accurately,
you will cut the length from
negative one to negative two
into four equally sized parts.
Each part has a length of 1/4 of a unit.
We have zero, negative
one, negative 1 1/4,
negative 1 2/4, and negative 1 3/4.
Because negative two is listed first
and negative two is to the
left of negative 1 3/4,
negative two is less than negative 1 3/4.
And now let's make the comparison
by obtaining a common denominator.
You will write negative two as a fraction
with a denominator of one.
You will also convert negative 1 3/4
to an improper fraction.
The improper fraction
is going to be negative.
The denominator remains four,
and the numerator is four
times one plus three,
which is seven.
Negative 1 3/4 equals negative 7/4.
The least common denominator is four,
which means we need to
write negative two over one
as an equivalent fraction
with a denominator of four.
We multiply the numerator
and denominator by four.
This gives us negative 8/4
compared to negative 7/4.
Now that we have a common denominator,
we can compare the fractions
by simply comparing the numerators.
Because negative eight is
less than negative seven,
we know negative two is
less than negative 1 3/4.
Let's look at the last two
examples on the next slide.
For C, we want to compare
negative 3/5 and negative 4/5,
which are both located
between zero and negative one.
And because both denominators are five,
we will cut the length
from zero to negative one
into five equal sized parts.
Each small length has a
length of 1/5 of a unit.
Negative 3/5 is 3/5 of a
unit to the left of zero,
starting at zero we have
negative 1/5, negative 2/5,
negative 3/5 is located here,
and one more fifth of a unit
left gives us negative 4/5.
Because we are starting with negative 3/5
and negative 3/5 is to
the right of negative 4/5,
negative 3/5 is greater than negative 4/5.
Notice how we already
have a common denominator
so to compare negative
3/5 and negative 4/5.
With the negative sign in the numerator,
we can compare the fractions
by simply comparing the numerators again,
because we have a common denominator.
Negative three is greater
than negative four,
which tells us that negative 3/5
is greater than negative 4/5.
And for the last example,
we want to compare negative
four and negative 11/3.
Negative four is four
units to the left of zero,
located here.
To plot negative 11/3 on the number line,
let's write negative
11/3 as a mixed number.
We know it's going to be
a negative mixed number
and to find the mixed number,
you will divide 11 by three.
There are three threes in 11.
Three times three is nine.
We subtract, the difference is two,
which means the quotient is three and 2/3,
giving a mixed number of negative 3 2/3
for negative 11/3.
Remember, this fraction here is formed
by writing the remainder over the divisor.
Negative 3 2/3 is 3 2/3
units to the left of zero,
which falls between negative
three and negative four.
Because the denominator is three,
we will cut the length from
negative three to negative four
into three equal sized parts.
Each of these small lengths
has a length of 1/3 of a unit.
So we have zero, negative one,
negative two, negative three,
negative 3 1/3, negative 3 2/3.
We start with negative four here
and negative four is to
the left of negative 3 2/3,
which means that negative four
is less than negative 3 2/3.
Let's also make the comparison
by obtaining a common denominator.
We will write negative four as a fraction
with a denominator of one
and then we'll leave negative
11/3 as an improper fraction
and obtain a common
denominator, which is three.
We multiply the numerator and denominator
of negative 4/1 by three.
This gives us the comparison of
negative 12/3 to negative 11/3.
Again, because we have
a common denominator,
we can make the comparison
by simply comparing the numerators.
Again, as long as the negative
sign is in the numerator.
And because negative 12
is less than negative 11,
we know negative four is
less than negative 11/3.
I hope you found this helpful.
