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.