INEQUALITIES – REAL NUMBERS

Concept: In an inequality, two unequal numbers are compared to determine which is larger and which is smaller. Follow the Directives below to make that determination.

Use the following symbols to express number comparisons.

= | means equals | 234 = 234 | 234 equals 234
> | means greater than | 55 > 3 2 | 55 is greater than 32
< | means less than | 23 < 66 | 23 is less than 66

HINT: Consider the symbols for is greater than and is less than as arrows which always point to the smaller number.

Real Number Directives

Directive 1 | When numbers have both whole number and decimal parts, compare the whole number parts first, beginning on the left and moving right. If the whole number parts have the same number of digits, then compare those digits beginning on the left and moving right.
Example 1 | Compare 54.321 and 53.412. |
Each number contains both whole number and decimal parts. Compare the whole number parts first.
Step 1 | Count the number of digits in the whole number part of the first number. There are two digits in the whole number part of the first whole number. | 5 4. 3 2 1 1 2
Step 2 | Count the number of digits in the whole number part of the second number. There are two digits in the whole number part of the second whole number. | 5 3. 4 1 2 1 2
Step 3 | Compare the number of digits in the whole number parts of both numbers. There are two digits in the whole number part of the first number (54.321) and two digits in the whole number part of the second number (53.412). | 5 4. 3 2 1 5 3. 4 1 2 1 2
Step 4 | Since both numbers have the same number of digits in the whole number part, compare the values of the whole number digits from left to right. The leftmost digit in both numbers is the same, 5. | 5 4.321 5 3.412
| Look one digit to the right of 5 and compare. The digit to the right of 5 in the number 5 4. 3 2 1 is 4. The digit to the right of 5 in the number 5 3. 4 1 2 is 3. 4 is greater than 3. | 5 4.321 5 3.412
| Therefore, 5 4. 3 2 1 is greater than 5 3. 4 1 2 | 54.321 >53.412

Directive 2 | When numbers have both whole number and decimal parts, compare the whole number parts first, beginning on the left and moving right. If the whole number parts do not have the same number of digits, then the number with the most digits in the whole number part always has the greater value.
Example 2 | Compare 654.32 and 76.54.
| Each number contains both whole number and decimal parts. Compare the whole number parts first.
Step 1 | Count the number of digits in the whole number part of the first number. There are three digits in the whole number part of the first whole number. | 6 5 4 . 32 1 2 3
Step 2 | Count the number of digits in the whole number part of the second number. There are two digits in the whole number part of the second whole number. | 7 6 . 54 1 2
Step 3 | Compare the number of digits in the whole number parts of both numbers. There are three digits in the whole number part of 654.32 and two digits in the whole number part of 76.54. | 6 5 4. 32 1 2 3 7 6 . 54 1 2
| Therefore, 654.32 is greater than 76.54. | 654.32 >76.54

Directive 3 | When numbers have both whole number and decimal parts, compare the whole number parts first, beginning on the left and moving right. If the whole number parts are the same (equal in number of digits and value), then compare the decimal parts, moving from left to right and comparing values.
Example 3 | Compare 65.765 and 65.732.
| Compare the whole number parts first, beginning on the left and moving right.
Step 1 | Count the number of digits in the whole number part of the first number. There are two digits in the whole number part of the first whole number. | 6 5 . 765 1 2
Step 2 | Count the number of digits in the whole number part of the second number. There are two digits in the whole number part of the second whole number. | 6 5 . 732 1 2
Step 3 | Compare the number of digits in the whole number parts of both numbers. There are two digits in the whole number part of the first number (65.765) and two digits in the whole number part of the second number (65.732). | 6 5 . 765 1 2 6 5 . 732 1 2
Step 4 | Since both numbers have the same number of digits in the whole number part, compare the values of the whole number digits from left to right. The leftmost digit in both numbers is the same, 6. | 6 5.765 6 5.732
| Look one digit to the right of 6 and compare. The digit to the right of 6 in the number 65.765 is 5. The digit to the right of 6 in the number 65.732 is 5. | 6 5.765 6 5.732
| The whole number parts are the same (equal in number of digits and value) in both 65.765 and 65.732. Therefore, compare the decimal parts, moving from left to right and comparing values.
Step 5 | Look one digit to the right of the decimal point in the first number, 65.765. The digit to the right of the decimal point in the first number is 7. | 6 5 . 7 6 5
| Look one digit to the right of the decimal point in the second number, 65.732. The digit to the right of the decimal point in the second number is also 7. | 6 5 . 7 3 2
Step 6 | Since the digits to the right of the decimal point in both numbers is the same, look one digit to the right of that digit. In 6 5.765 the digit to the right of 7 is 6. In 6 5.732 the digit to the right of 7 is 3. 6 is greater than 3. | 6 5 . 7 6 5 6 5 . 7 3 2
| Therefore, 6 5.7 65 is greater than 6 5.732. | 6 5.7 65>6 5.732

Directive 4 | When dealing with signed numbers, the number farthest to the left on a number line is the smaller number.

Example Compare – 8 and – 4. Since – 8 is further to the left on the number line than – 4 is, – 8 < – 4.