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May 23rd

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# Rational Numbers

A rational number is a number that can be expressed in the form , where a and b are integers and b is not equal to 0.

 Comparing Rational Numbers If the graph of a is to the left of the graph of b on a number line, then a < b. For any two numbers a and b, exactly one of these is true: a < b, a = b, a > b You can use cross products to compare two fractions. For rational numbers and , with b >0 and d > 0, if , then ad < bc, and if ad < bc, then .

Example

Use a calculator to write the fractions as decimals. Then order the fractions from least to greatest.

Solution

The decimals from least to greatest are So the fractions should be ordered .

## Adding and Subtracting Rational Numbers

To add or subtract rational numbers, use the same rules you used to add integers. When you are adding three or more rational numbers, you can use the commutative and associative properties to rearrange the addends.

Example

Find .

Solution

The LCD is 8.

The sum is negative.

Subtract absolute values.

## Multiplying Rational Numbers

The product of two numbers having the same sign is positive. The product of two numbers having different signs is negative. It is also useful to note that multiplying a number or expression by -1 results in the opposite of the number or expression. This is called the multiplicative property of -1.

Example

Evaluate

 Replace x with Divide out common factors. Multiply. The signs are different, so the product is negative.

## Dividing Rational Numbers

You can use the same rules of signs when dividing rational numbers that you used for multiplying.

 Dividing Two Rational Numbers The quotient of two numbers having the same sign is positive.The quotient of two numbers having different signs is negative.

If a fraction has one or more fractions in the numerator or denominator, it is a complex fraction. To simplify a complex fraction, rewrite it as a division expression.

Example

Simplify

Solution

Rewrite the complex fraction as

 Multiply by the reciprocal of -8. The signs are different, so the product is negative.
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