Algebra I. Lesson 2

Instruction 2-5

 Pre-Test Post-Test Q&A Q&A

Integers and the Number Line | Adding and Subtracting Integers | Inequalities and the Number Line | Comparing and Ordering Rational Numbers | Dividing Rational Numbers | Summary

Dividing Rational Numbers

First we need to know what the reciprocal of a rational number is. If the numerator and denominator of a rational number are replaced by each other, the new number is called the reciprocal of it.

That is, for two integers a and b different from zero, is the reciprocal of .

For example, – is the reciprocal of and is the reciprocal of .

If you have a whole number, such as 2, it can be written as .

The reciprocal of would be .
Practice 18. Complete Table 2.16.

Rational Number Numerator Denominator Reciprocal

Table 2.16

Solution.

Rational Number Numerator Denominator Reciprocal

12

17

–7

16

6

–5

–11

–23

Table 2.17

Dividing two rational numbers is very easy. When dividing a rational number by the second number, simply find the reciprocal of the second number. Then multiply the first number by the reciprocal. For example, let try ÷. The reciprocal of is .

Multiply by

÷ = ×

= = = 2.

In dividing rational numbers, we use the following rules for the division of signs.

Case 1. If two rational numbers of like signs are divided, the sign of the result is always positive. For example, the results of (+) ÷ (+) and (–) ÷ (–) are both positive.

Case 2. If two rational numbers of unlike signs are divided, the result is always negative.

For example, the results of (–) ÷ (+) and (+) ÷ (–) are both negative.

Practice 19. Solve.
(a) – ÷ (b) (–) ÷ (–)(c) ÷ (–)

Solution.
(a) – ÷ = – ×

= –

= –

(b) (–) ÷ (–) = (–) × (–)

=
=
=

(c) ÷ (–) = – ×

= –
= –
= –

Real Life Application 6.

The area of a lot shown in Figure 2.18 is 240 ft2. The width of the lot is feet. The formula for the length of a rectangle is: , where L = length, A = area and W = width.

Find the measure of its length.

Figure 2.18

Solution. The length

Length of Rectangle =
Length of the Lot = 240 ÷
= ×

=
=
= feet

If you have a graphing calculator, TI82, TI83 or TI84, there is an alternative method that can be used to divide fractions. If you have divided by , enter the following commands into the calculator: {(172 35) (86 70) math enter enter} . and you will have the answer in the form of a fraction. Be especially careful to put each fraction inside its own set of parentheses, or you will get an extremely wrong answer. A nice little bonus of this method is that the fraction is reduced to its lowest terms. (In fact, if you need to reduce a fraction to lowest terms, such as , you can put the fraction into the calculator and punch the buttons math, then enter, then enter once again and the fraction will be reduced to lowest terms. In this case, the lowest term is.

Practical Exercise 4 Compute

(a) ÷
(b) ÷
(c) ÷

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Now let's do Practice Exercise 2-5 (top).

Summary

Integers.

The set of integers is {, … , –3, –2, –1, 0, 1, 2, 3, …, }.

Number Line.

A line that graphically expresses the real numbers as a series of points distributed about a point arbitrarily designated as zero and in which the magnitude of each number is represented by the distance of the corresponding point from zero. The positive numbers are to the right of zero and the negative numbers are to the left of zero. This last statement is mine.

Absolute Value.

The distance of any rational number from zero is called its absolute value.

All the integers are of like signs. Add the absolute values of the numbers. Then choose the sign of the numbers as the sign of the result.

Two integers are of unlike signs. Subtract the smaller absolute value from the larger one. Determine the sign of the number with larger absolute value. Use this sign as the sign of the result.

General Rules for Combing Integers.

If a and b are integers, then

a + (–b) = ab

a – (+b) = ab

a – (–b) = a + b

a + (+b) = a + b

Inequalities and the Number Line.

We use a number line to compare numbers.

On a number line, all the numbers to the right of a number are greater than that number.

All the numbers to the left of a number are smaller than that number.

What are Rational Numbers?

By dividing each pair of integers, we can define a new set of numbers. This is called the set of rational numbers. If there are two integers, a and b and b≠ 0 are two integers,

then a number is called a rational number.

Comparing and Ordering Rational Numbers.

A good way to compare two rational numbers is by graphing them on a number line. On a number line, the number to the right of another number is greater.

Dividing Rational Numbers.

When dividing a rational number by a second rational number, simply find the reciprocal of the second number. Then multiply the first number by the reciprocal of the second number.

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