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The absolute value operation is distributive with respect to multiplication. This means that the absolute value of a product equals the product of the absolute values.

To demonstrate, consider the following expression:

We can evaluate this expression in two separate ways:

The first is to find the absolute value of the product. In doing this, we get

The second is to find the product of the absolute values. In doing this, we get

In general

The absolute value operation is also distributive with respect to division. In general,

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Example: Distributing the Absolute Value Over Multiplication

Simplify the expression

EXPLANATION

Firstly, since we can write our expression as

Then, we distribute the absolute value over the multiplication, and simplify:

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Practice: Distributing the Absolute Value Over Multiplication

$|5x|=$

a	$-\|5x\|$
b	$-5\|x\|$
c	$-5$
d	$5$
e	$5\|x\|$

EXPLANATION

We distribute the absolute value over the multiplication, and simplify:

$\begin{align*} |5x| &= \\[3pt] |5 \cdot x| &= \\[3pt] |5| \cdot |x| &= \\[3pt] 5 \cdot |x| &= \\[3pt] 5|x| \end{align*}$

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Practice: Distributing the Absolute Value Over Multiplication

Simplify the expression $|-11x|.$

a	$-\|11x\|$
b	$11\|x\|$
c	$-11\|x\|$
d	$11$
e	$-11$

EXPLANATION

Firstly, since $|-a| = |a|,$ we can write our expression as

$|11x|.$

Then, we distribute the absolute value over the multiplication, and simplify:

$\begin{align*} |11x| &=\\[5pt] |11\cdot x| &=\\[5pt] |11| \cdot |x| &= \\[5pt] 11 \cdot |x| &= \\[5pt] 11|x| \end{align*}$

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Example: Distributing the Absolute Value Over Division

Simplify the expression

EXPLANATION

Firstly, since we can write our expression as

Then, we distribute the absolute value over the division, and simplify:

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Practice: Distributing the Absolute Value Over Division

$\left|-\dfrac{3}{2x}\right|=$

a	$\dfrac{3}{2}$
b	$-\dfrac{3}{2\|x\|}$
c	$-6x$
d	$\dfrac{3}{2\|x\|}$
e	$6x$

EXPLANATION

Firstly, since $|-a| = |a|,$ we can write our expression as

$\left|\dfrac{3}{2x}\right|.$

Then, we distribute the absolute value over the division, and simplify:

$\begin{align*} \left|\dfrac{3}{2x}\right| &=\\[5pt] \dfrac{|3|}{|2x|} &= \\[5pt] \dfrac{3}{|2x|} &= \\[5pt] \dfrac{3}{|2 \cdot x|} &= \\[5pt] \dfrac{3}{|2| \cdot |x|} &= \\[5pt] \dfrac{3}{2|x|} \end{align*}$

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Practice: Distributing the Absolute Value Over Division

Simplify the expression $\left|-\dfrac{5x}{3y}\right|.$

a	$-\dfrac{5\|x\|}{3\|y\|}$
b	$\dfrac{5\|x\|}{3\|y\|}$
c	$\dfrac{5}{3}$
d	$15xy$
e	$-15xy$

EXPLANATION

Firstly, since $|-a| = |a|,$ we can write our expression as

$\left|\dfrac{5x}{3y}\right|.$

Then, we distribute the absolute value over the division, and simplify:

$\begin{align*} \left|\dfrac{5x}{3y}\right| &=\\[5pt] \dfrac{|5x|}{|3y|} &= \\[5pt] \dfrac{|5 \cdot x|}{|3 \cdot y|} &= \\[5pt] \dfrac{|5| \cdot |x|}{|3| \cdot |y|} &= \\[5pt] \dfrac{5|x|}{3|y|} \end{align*}$

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CANCEL

a	$\left\| \dfrac{4z}{7} \right\|$
b	$\left\| \dfrac{4}{7} \right\|$
c	$- \dfrac{4}{7}$
d	$\dfrac{4}{7}$
e	$-\left\| \dfrac{4z}{7} \right\|$

a	$-\|5x\|$
b	$-5\|x\|$
c	$-5$
d	$5$
e	$5\|x\|$

a	$-\|11x\|$
b	$11\|x\|$
c	$-11\|x\|$
d	$11$
e	$-11$