Absolute Value: Practice

Work the bars first: $\lvert -4 \rvert = 4$. The minus sign sits outside the bars, so it now applies to that result.

$-\lvert -4 \rvert = -(4) = -4$

The bars measure only what stands between them, so the minus in front was never part of the measuring. The result is the opposite of an absolute value, which is allowed to be negative.

The bars group the product, so multiply inside first. A positive times a negative is negative, giving $-12$, and then the bars measure its distance from zero.

$\lvert 3 \times (-4) \rvert = \lvert -12 \rvert = 12$

Add inside the bars first. Adding two negatives moves further left: $-2$ and $-5$ combine to $-7$.

$\lvert (-2) + (-5) \rvert = \lvert -7 \rvert = 7$

Each pair of bars is its own group. Both equal $10$.

$\lvert -10 \rvert - \lvert -10 \rvert = 10 - 10 = 0$

The size of a change is the distance between the two readings, which is the absolute value of their difference.

$\lvert 9 - (-6) \rvert = \lvert 9 + 6 \rvert = \lvert 15 \rvert = 15$

The temperature changed by $15$ degrees.

Work from the innermost grouping outward. Inside the parentheses, $3 - 8 = -5$, then multiply by $2$ to get $2 \times (-5) = -10$. Finally the bars measure its distance from zero.

$\lvert 2 \times (3 - 8) \rvert = \lvert -10 \rvert = 10$

Each pair of bars is its own group, so take each absolute value first, then multiply.

$\lvert -3 \rvert \times \lvert -4 \rvert = 3 \times 4 = 12$

Absolute value is a distance, so it is always zero or positive, for every integer.

$\lvert x \rvert \ge 0$

The choice $\lvert x \rvert = x$ fails for negatives, $\lvert x \rvert > 0$ fails at $x = 0$, and $\lvert x \rvert = -x$ fails for positives, so only $\lvert x \rvert \ge 0$ holds for all $x$.

The bars group the quotient, so divide inside first. A negative divided by a positive is negative, giving $-3$, and then the bars measure its distance from zero.

$\lvert (-12) \div 4 \rvert = \lvert -3 \rvert = 3$

Evaluate inside each pair of bars first. The first is $\lvert -2 - 3 \rvert = \lvert -5 \rvert = 5$, and the second is $\lvert 1 - 1 \rvert = \lvert 0 \rvert = 0$.

$5 + 0 = 5$

Work the bars first: $6 - 13 = -7$, so $\lvert 6 - 13 \rvert = 7$. The minus sign outside then applies to that result.

$-\lvert 6 - 13 \rvert = -(7) = -7$

Two integers are eight units from zero: $8$ and $-8$. The condition that $x$ is negative picks out one of them.

$\lvert -8 \rvert = 8 \text{ and } -8 < 0$

So $x = -8$. The extra condition removes the ambiguity that $\lvert x \rvert = 8$ alone would leave.