INEQUALITIES

SIGNED DECIMAL 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

Directive 1 | When two positive decimals have the same number of digits, compare the digits in each decimal 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 | Count the number of digits in the first decimal. There are three digits in 0.345. | 0.3 4 5 1 2 3
Step 2 | Count the number of digits in the second decimal. There are three digits in 0.323. | 0.3 2 3 1 2 3
Step 3 | Compare the digits in each decimal 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 less than 0.3 4 5. | 0.323 < 0.345

Directive 2 | When two negative decimals have the same number of digits, compare the digits in each decimal 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 | Count the number of digits in the first decimal. There are three digits in 0.345. | 0.3 4 5 1 2 3
Step 3 | Count the number of digits in the second decimal. There are three digits in 0.323. | 0.3 2 3 1 2 3
Step 4 | Compare the digits in each decimal 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 5 | Re-apply the negative signs on the numbers. | -0.323 -0.345
Step 6 | Remember that when comparing negative decimals, the LARGER number has the SMALLER value. | -0.323 > -0.345
Directive 3 | When two positive decimals have a different number of digits, add zeros to the right of the decimal with fewer digits until the two decimals have the same number of digits. Then compare the digits in each decimal from left to right to determine which decimal is the larger number. |  |
Example | Compare 0.354657 and 0.353. | 0.354657 0.353 |
Step 1 | Count the number of digits in the first decimal. There are six digits in 0.354657 | .3 5 4 6 5 7 1 2 3 4 5 6 |
Step 2 | Count the number of digits in the second decimal. There are three digits in 0.353 | .3 5 3 1 2 3 |
Step 3 | Add zeros to the right of the decimal with fewer digits until the two decimals have the same number of digits. Since 0.353 has fewer digits than 0.354657, add zeros to the right of 0.353. | .3 5 3 0 0 0 |
Step 4 | Compare the digits in each decimal 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 53000 |
| 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 3000 is 5. | 0.3 5 4657 0.3 5 3000 |
| 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 3 000 is 3. 3 is less than 4. | 0.3 5 4 657 0.3 5 3 000 |
| Therefore, 0.3 5 3 000 is less than 0.3 5 4 657. | 0.353000 < 0.354657 |

Directive 4 | When two negative decimals have a different number of digits, add zeros to the right of the decimal with fewer digits until the two decimals have the same number of digits. Then compare the digits in each decimal from left to right to determine which decimal is the larger number. With two 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 | Count the number of digits in the first decimal. There are six digits in 0.354657 | .3 5 4 6 5 7 1 2 3 4 5 6
Step 3 | Count the number of digits in the second decimal. There are three digits in 0.353 | .3 5 3 1 2 3
Step 4 | Add zeros to the right of the decimal with fewer digits until the two decimals have the same number of digits. Since 0.353 has fewer digits than 0.354657, add zeros to the right of 0.353. | .3 5 3 0 0 0
Step 5 | Compare the digits in each decimal 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 53000
| 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 3000 is 5. | 0.3 5 4657 0.3 5 3000
| 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 3 000 is 3. 3 is less than 4. | 0.3 5 4 657 0.3 5 3 000
| Therefore, 0.3 5 3 000 is less than 0.3 5 4 657. | 0.353000 < 0.354657
Step 6 | Re-apply the negative signs on the numbers. | -0.353000 -0.354657
Step 7 | Remember that when comparing negative decimals, the LARGER number has the SMALLER value. | -0.353000 > 0.354657

Directive 5 | 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.