Number Theory - Pre-Algebra

To find the distance on a number line:

The fraction is less than and greater than , so it must fall between those points on the number line. Negative numbers are to the left of while positive numbers are to the right.

Because is less than , the point must be closer to than .

The point lies halfway between and , or at .

= less than

= greater than

= less than or equal to

= greater than or equal to

Thus, in increasing order:

When looking at the number line, we can see that x is any integer between 2 and 7.

Anything to the right of an integer is considered "greater than" and anything left of an integer is considered "less than".

So, we know that x is greater than 2, but less than 7.

Now, since the circle above the 2 is shaded in, 2 is included in the solution. The circle above the 7 is not shaded in, so 7 is not included in the solution.

We now know that x is greater than or equal to 2, and it is still less than 7.

Using the statement above, we will create 2 inequalities.

Now, can also be written as . It has the same value, it just looks a little different.

So we have the two inequalities

and

We can combine them, and it'll look like this

When looking at the number line,

we see at -5 there is a filled in circle. We also see there is an arrow going to the right of -5. Every integer to the right of -5 is greater than -5 itself. Because the circle above -5 is filled in, we know -5 is included in the inequality. We get

There is only one negative number in the numbers.

The next biggest is ,

Next is ,

Next is ,

Lastly is

To find the distance on a number line:

The fraction is less than and greater than , so it must fall between those points on the number line. Negative numbers are to the left of while positive numbers are to the right.

Because is less than , the point must be closer to than .

The point lies halfway between and , or at .

= less than

= greater than

= less than or equal to

= greater than or equal to

Thus, in increasing order:

When looking at the number line, we can see that x is any integer between 2 and 7.

Anything to the right of an integer is considered "greater than" and anything left of an integer is considered "less than".

So, we know that x is greater than 2, but less than 7.

Now, since the circle above the 2 is shaded in, 2 is included in the solution. The circle above the 7 is not shaded in, so 7 is not included in the solution.

We now know that x is greater than or equal to 2, and it is still less than 7.

Using the statement above, we will create 2 inequalities.

Now, can also be written as . It has the same value, it just looks a little different.

So we have the two inequalities

and

We can combine them, and it'll look like this

When looking at the number line,

we see at -5 there is a filled in circle. We also see there is an arrow going to the right of -5. Every integer to the right of -5 is greater than -5 itself. Because the circle above -5 is filled in, we know -5 is included in the inequality. We get

There is only one negative number in the numbers.

The next biggest is ,

Next is ,

Next is ,

Lastly is

The sum (or difference) or 2 even integers is even. Similarly, the product (or quotient) of 2 even integers is also even; therefore the answer must be , which can be easily checked by plugging in any two even numbers.

For example, if , which is odd.

By definition, an integer is a whole number that is either positive or negative. Negative numbers such as and are integers, as are 10 and 5.

, however, is not an integer because it isn't a whole number.

The only prime factors are 3 and 5, therefore, 5 will be the greatest prime factor.

An integer is a whole number, either positive or negative. In other words, it is a number that can be written without a fractional or decimal part. Examples are:

is therefore not an integer, but it is a fraction.