INEQUALITIES

SIGNED WHOLE 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 whole numbers have opposite signs, the value of the positive whole number is always greater than the value of the negative whole number. |
Example | Compare -41 and 393. | -41 393
Step 1 | Determine the sign of the first number. The sign of the first number is negative. | -41
Step 2 | Determine the sign of the second number. The sign of the second number is positive. | 393
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 393 is greater than the negative number -41. | 393 > -41

Directive 2 | 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 3 | When two negative whole 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 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.