These questions ask you to evaluate absolute values, compare them, find integers at a stated distance, or continue with an operation after the bars are resolved. The safest habit is to mark exactly what each pair of bars encloses.
Resolve the bars before the outside work
Read absolute value as distance from zero. The distance of -11 from zero is 11, the distance of 4 is 4, and the distance of zero is zero. A positive distance usually corresponds to two integers, one on each side of zero; for example, the integers at distance 5 are -5 and 5.
| Expression shape | Decision method |
|---|---|
| One integer inside bars | Return its nonnegative distance from zero. The original side of zero no longer affects the result. |
| A calculation inside bars | Complete the enclosed calculation first, then take the distance of that result; the absolute value of -6 plus 10, enclosed together, is 4. |
| A sign outside the bars | Evaluate the bars first and then apply the outer sign; a minus outside the absolute value of -8 produces -8. |
| Further arithmetic outside | Replace each absolute value with its distance, then continue normally. The final result may be negative after subtraction or multiplication. |
Separate bars matter. The absolute values of -4 and 10, taken separately and added, give 14, while the absolute value of their sum gives 6. Always follow the visible grouping instead of moving bars across a plus or minus sign.
Traps around the bars
| Tempting route | How to reject it |
|---|---|
| Carry a negative sign through the bars, or flip every positive value | The bars report distance. They remove direction from a negative input and leave an already positive input unchanged. |
| Carry the order of negative values into their absolute values | Compare the returned magnitudes instead. A negative farther from zero has the larger absolute value even though the original integer is smaller. |
| Cancel an outer minus with the minus inside | They act at different stages. The inner sign helps determine the distance; the outer sign acts on the finished distance. |
| Apply bars to just one term of an enclosed sum | The bars govern all the content between them. Finish that content as one calculation before measuring its distance. |
| Name only the positive integer at a positive distance | Check both sides of zero. Two opposite integers share that same distance. |
| Stretch the bars over a later multiplication | Bars affect only what they visibly enclose. Resolve them, then use the ordinary sign rule for the remaining product. |
Try it
Enter the value of |-9|.
Work out |-4| + |6| and then |-4 + 6|. Which pair of values is correct, in that order?
- 10 and 2
- 2 and 10
- 10 and 10
- 2 and 2
Enter the value of |-7| - |-10|.