Absolute Value: Practice

Absolute value is the distance from zero. The number $6$ sits six units to the right of zero, and a number that is already positive is its own distance.

$\lvert 6 \rvert = 6$

The number itself is $-9$, but the bars report the distance, and the distance is nine units either way you walk it.

$\lvert -9 \rvert = 9$

The side of zero the number sits on does not enter the answer.

The number $-1$ is one unit to the left of zero, so its distance from zero is $1$.

$\lvert -1 \rvert = 1$

The number $-25$ lies twenty-five units to the left of zero.

$\lvert -25 \rvert = 25$

Absolute value is a distance, and a distance is a count of equal steps along the line, which cannot be less than zero.

$\lvert x \rvert \ge 0 \text{ for every number } x$

So an absolute value can be zero or positive, but never negative.

The number $-8$ is eight units to the left of zero.

$\lvert -8 \rvert = 8$

The number $-7$ is seven units from zero, so its absolute value is the positive distance $7$.

$\lvert -7 \rvert = 7$

The bars report a distance, and a distance is never negative, so the answer is $7$ and not $-7$.

Zero sits right on zero, so there are no steps to count.

$\lvert 0 \rvert = 0$

Zero is neither to the left of zero nor to the right of it, but it still has a distance from zero, and that distance is $0$.

Replace each with its distance from zero, then compare.

$\lvert -5 \rvert = 5, \quad \lvert 2 \rvert = 2$

Since $5 > 2$, the value $\lvert -5 \rvert$ is larger.

The number $-2$ is two units to the left of zero.

$\lvert -2 \rvert = 2$

Walk four units from zero. There are two directions to walk, so there are two places to land.

$\lvert 4 \rvert = 4 \quad\text{and}\quad \lvert -4 \rvert = 4$

The two are opposites, which is why they tie: a mirror across zero changes the side without changing how far out a number sits.

The number $-3$ is three units to the left of zero.

$\lvert -3 \rvert = 3$