INEQUALITIES

SIGNED 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 help 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

Signed Real Number Directives

Directive 1 | 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.

Directive 2 | When two numbers have opposite signs, the value of the positive number is always greater than the value of the negative number.
Example | Compare -33.7 and 1.7. | -33.7 1.7
Step 1 | Determine the sign of the first number. The sign of the first number is negative. | -33.7
Step 2 | Determine the sign of the second number. The sign of the second number is positive. | 1.7
Step 3 | When two whole numbers have opposite signs, the value of the positive whole number is always greater than the value of the negative whole number. The positive number 1.7 is greater than the negative number –33.7. | 1.7 > -33.7

Directive 3 | When two negative whole numbers do not have the same number of digits, the number with the most digits has the smaller value.
Example | Compare -8674 and -567. | -8674 -567
Step 1 | Disregard the negative signs on the numbers. | 8674 567
Step 2 | Count the number of digits in the first number. There are four digits in the first number. | 8 6 7 4 1 2 3 4
Step 3 | Count the number of digits in the second number. There are three digits in the second number. | 5 6 7 1 2 3
Step 4 | Compare the number of digits. Since there are four digits in the first number (8674) and three digits in the second number (567), the first number is greater than the second number. | 8674 > 567
Step 5 | Re-apply the negative signs on the numbers. | -8674 -567
Step 6 | Remember that when two negative whole numbers do not have the same number of digits, the number with the most digits has the smaller value. | -8674 < -567

Directive 4 | When two negative numbers have the same number of digits, compare the digits from left to right to determine which number is larger. With two negative numbers, the larger number always has the smaller value.
Example | Compare -567 and -578. | -567 -578
Step 1 | Disregard the negative signs on the numbers. | 567 578
Step 2 | Count the number of digits in the first number. There are three digits in 567. | 5 6 7 1 2 3
Step 3 | Count the number of digits in the second number. There are three digits in 578. | 5 7 8 1 2 3
Step 4 | Compare the number of digits. There are three digits in the first number (567) and three digits in the second number (578). | 5 6 7 5 7 8 1 2 3
Step 5 | Since both numbers have the same number of digits, compare the digits from left to right. The leftmost digit in both numbers is the same, 5. | 5 67 5 78
| Look one digit to the right of 5 and compare. The digit to the right of 5 in the number 5 6 7 is 6. The digit to the right of 5 in the number 5 7 8 is 7. 6 is less than 7. | 5 6 7 5 7 8
| Therefore, 567 is less than 578. | 567 < 578
Step 6 | Re-apply the negative signs on the numbers. | -567 -578
Step 7 | Remember that with two negative numbers, the larger number always has the smaller value. | -567 > -578

Directive 5 | When two positive decimals have the same number of digits, compare the digits in both decimals, moving from left to right, to determine which decimal is the larger number.
Example | Compare 0.345 and 0.323. | 0.345 0.323
Step 1 | Compare the digits in both decimals, moving from left to right, to determine which decimal is greater. The leftmost digit in both numbers is the same, 3. | 0.3 45 0.3 23
Step 2 | Look one digit to the right of 3 and compare. The digit to the right of 3 in the number 0.3 4 5 is 4. The digit to the right of 3 in the number 0.3 2 3 is 2. 2 is less than 4. | 0.3 4 5 0.3 2 3
Step 3 | Therefore, 0.3 2 3 is less than 0.3 4 5. | 0.323 < 0.345

Directive 6 | When two negative decimals have the same number of digits, compare the digits in both decimals, moving from left to right, to determine which decimal is the larger number. When comparing negative decimals, the LARGER number has the SMALLER value.
Example | Compare - 0.345 and - 0.323. | - 0.345 - 0.323
Step 1 | Disregard the negative signs on the numbers. | 0.345 0.323
Step 2 | Compare the digits in both decimals, moving from left to right, to determine which decimal is greater. The leftmost digit in both numbers is the same, 3. | 0.3 45 0.3 23
| Look one digit to the right of 3 and compare. The digit to the right of 3 in the number 0.3 4 5 is 4. The digit to the right of 3 in the number 0.3 2 3 is 2. 2 is less than 4. | 0.3 4 5 0.3 2 3
| Therefore, 0.3 2 3 is smaller than 0.3 4 5. | 0.323 < 0.345
Step 3 | Re-apply the negative signs on the numbers. | - 0.323 - 0.345
Step 4 | Remember that when comparing negative decimals, the LARGER number has the SMALLER value. | - 0.323 > - 0.345

Directive 7 | When two negative decimals have a different number of digits, compare the digits in both decimals, moving from left to right, to determine which decimal is the larger number. When comparing negative decimals, the LARGER number has the SMALLER value.
Example | Compare - 0.354657 and - 0.353. | - 0.354657 - 0.353
Step 1 | Disregard the negative signs on the numbers. | 0.354657 0.353
Step 2 | Compare the digits in both decimals, moving from left to right, to determine which decimal is larger. The leftmost digit in both numbers is the same, 3. | 0.3 54657 0.3 53
| Look one digit to the right of 3 and compare. The digit to the right of 3 in the number 0.3 5 4657 is 5. The digit to the right of 3 in the number 0.3 5 3 is 5. | 0.3 5 4657 0.3 5 3
| Look one digit to the right of 5 and compare. The digit to the right of 5 in the number 0.3 5 4 657 is 4. The digit to the right of 5 in the number 0.3 5 3 is 3. 3 is less than 4. | 0.3 5 4 657 0.3 5 3
| Therefore, 0.3 5 3 is less than 0.3 5 4 657. | 0.353 < 0.354657
Step 6 | Re-apply the negative signs on the numbers. | - 0.353 - 0.354657
Step 7 | Remember that when comparing negative decimals, the LARGER number has the SMALLER value. | - 0.353 > - 0.354657

Directive 8 | When positive 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 | 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
Step 5 | Therefore, 5 4. 3 2 1 is greater than 5 3. 4 1 2 | 54.321 >53.412
Directive 9 | When negative 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. When comparing negative numbers, the LARGER number has the SMALLER value.
Example | Compare -54.321 and -53.412. |
Step 1 | Disregard the negative signs on the numbers. | 54.321 53.412
Step 2 | 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 3 | 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 4 | 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 5 | 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
Step 6 | Re-apply the negative signs on the numbers. | -54.321 -53.412
Step 7 | Remember that when comparing negative decimals, the LARGER number has the SMALLER value. | -54.321 < -53.412

Directive 10 | When positive 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 | 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
Step 4 | Therefore, 654.32 is greater than 76.54. | 654.32 >76.54

Directive 11 | When negative 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 smaller value. |
Example | 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
Step 4 | When comparing negative numbers, the number with the most digits in the whole number part always has the smaller value. Therefore, - 654.32 is smaller than -76.54. | - 654.32 < - 76.54

Directive 12 | When positive 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 | 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 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
Step 7 | Therefore, 6 5.7 65 is greater than 6 5.732. | 6 5.7 65 >6 5.732

Directive 13 | When negative 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 | 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 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
Step 7 | Remember that when comparing negative decimals, the LARGER number has the SMALLER value. Therefore, - 6 5.7 65 is smaller than - 6 5.732. | -6 5.7 65 < -6 5.732