The absolute value of a number is its numerical value when its sign is not considered.
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The absolute value of |125| is |125.|
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The absolute value of |-18| is |18.|
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The absolute value of |-\dfrac{3}{5}| is |\dfrac{3}{5}.|
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The absolute value of |-10.8| is |10.8.|
The absolute value of a number is always positive.
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If a number is positive, its absolute value is equal to the number itself.
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If a number is negative, its absolute value is equal to its opposite.
Qu’en est-il lorsqu’on fait des opérations arithmétiques avec des valeurs absolues? C’est ce qu’on vérifie dans les 2 prochains exemples.
Is the absolute value of |-5+3| equal to the sum of the absolute values of |-5| and |3|?
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First, we add |-5+3| which is |-2.| We then determine that the absolute value of |-2| is |\boldsymbol{2}.|
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The absolute value of |-5| is |5| and that of |3| is |3.| The, we perform the operation |5+3,| which is |\boldsymbol{8}.|
Answer: The absolute value of |-5+3| is not equal to the sum of the absolute values of |-5| and |3.|
Find the opposite of the absolute value of | -2.5| and the absolute value of the opposite of |-2.5.|
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The absolute value of |-2.5| is |2.5.| The opposite of |2.5| is |\boldsymbol{-2.5}.|
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The opposite of |-2.5| is |2.5.| The absolute value of |2.5| is |\boldsymbol{2.5}.|
If we determine the absolute value of numbers before performing a mathematical operation, we do not necessarily obtain the same result as if we determine it after the operation. The absolute value of the sum of 2 numbers is not always equal to the sum of the absolute value of the 2 numbers. It is the same for subtraction and this relates to other concepts such as opposite numbers.
It is therefore necessary to respect the order of operations when working with absolute values. These have the same priority as brackets.
The absolute value of a number allows us to consider the number without taking into account its sign.
In other words, if a number |x| is positive, then the absolute value of |x| is |x,| but if |x| is negative, then the absolute value of |x| is its opposite, which is |-x.| This is, in fact, the definition of the absolute value provided below.
The absolute value of a real number |x,| denoted |\vert x \vert,| is the following:
||\vert x \vert = \begin{cases}\ \ \ x &\text{if} &x \geq 0\\-x &\text{if } &x < 0\end{cases}||
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|\vert 125\vert = 125|
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|\vert\! -18\vert = 18|
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|\left\vert -\dfrac{3}{5}\right\vert = \dfrac{3}{5}|
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|-\vert\!-6\vert=-6|
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|\vert\! -2\times 45\vert = \vert\!-90\vert = 90|
This definition is used to solve equations containing an absolute value. For example, the equation |\vert x\vert = y| means that |x=y| or that |x=-y.| In other words, there are always 2 solutions to consider when solving such an equation.
From the definition of absolute values, we can deduce properties that are important to know when working with the absolute value function.
| Property | Example |
|---|---|
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The restriction on absolute values |
|\vert\! -1.3\vert =1.3| and |1.3\geq 0| |
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The absolute value of opposite numbers||\vert x\vert = \vert\! -x\vert|| |
||\begin{align}\vert 4.5\vert &= \vert\! -4.5\vert \\ 4.5&=4.5\end{align}|| |
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The absolute value of a product||\vert x\times y\vert = \vert x\vert \times \vert y\vert|| |
||\begin{align}\vert\! -2\times 6.4\vert &= \vert\! -2\vert \times \vert 6.4\vert \\ \vert\! -12.8\vert &= 2\times 6.4\\ 12.8 &= 12.8\end{align}|| |
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The absolute value of a quotient |\left\vert \dfrac{x}{y}\right\vert =\dfrac{\vert x \vert}{\vert y\vert}| if |y\neq 0| |
||\begin{align}\left\vert\dfrac{-12.3}{2}\right\vert &= \dfrac{\vert\! -12.3\vert}{\vert 2\vert}\\ \vert\! -6.15\vert &= \dfrac{12.3}{2}\\ 6.15 &= 6.15\end{align}|| |
The last 2 properties show that the absolute value of a product (or quotient) of numbers is equal to the product (or quotient) of the absolute values of those numbers. However, this does not apply to addition or subtraction. Here are some examples.
||\begin{align}\vert\!-5 + 2\vert &\overset{?}{=} \vert\!-5\vert + \vert 2\vert\\ \vert\!-3 \vert &\overset{?}{=} 5 + 2\\3&\color{#ec0000}{\boldsymbol\neq}7 \end{align}||
||\begin{align}\vert\!-4 + -6\vert &\overset{?}{=} \vert\!-4\vert + \vert\! -6\vert\\ \vert\!-10 \vert &\overset{?}{=} 4 + 6\\10&\color{#3a9a38}{\boldsymbol=}10 \end{align}||
||\begin{align}\vert\!-10 -6\vert &\overset{?}{=} \vert\!-10\vert - \vert6\vert\\ \vert\!-16 \vert &\overset{?}{=} 10 - 6\\16&\color{#ec0000}{\boldsymbol\neq}4 \end{align}||
In fact, the absolute value of a sum of numbers is always smaller than or equal to the sum of the absolute values of these numbers. This property, called the triangle inequality, translates into the following inequality: ||\vert x + y\vert \leq \vert x\vert + \vert y\vert||
For subtraction, we have this property: ||\vert x - y\vert \geq \big\vert\vert x\vert - \vert y\vert\big\vert||
The first property, which is the restriction on the absolute value, is used to check if an equation containing an absolute value has a valid solution or not.
For example, |\vert x \vert=-6| is an equation with no solution since it does not respect the restriction |\vert x\vert \geq 0.| In fact, the absolute value of any real number |x| is never a negative number.
The following examples show how to simplify algebraic expressions with an absolute value.
Note: It is important to respect the order of operations. The application of an absolute value has the same priority as brackets. Also, it is only possible to add, subtract or simplify absolute values if they are identical.
Example 1
In this example, we apply the property of the absolute value of a product to simplify the expression.
||\begin{align}3\vert\!\color{#3b87cd}{-5}\color{#3a9a38}{(x-4)}\vert&=3\times\color{#3b87cd}{\vert\!-5\vert}\times\color{#3a9a38}{\vert x-4\vert}\\&=3\times5\times\vert x-4\vert\\&=15\vert x-4\vert\end{align}||
Example 2
Here, we apply the property of the absolute value of a product, and then add the like terms to simplify the expression.
||\begin{align}\vert-2x+10\vert-6\vert x-5\vert&=\vert\color{#3b87cd}{-2}\color{#3a9a38}{(x-5)}\vert-6\vert x-5\vert\\&=\vert\color{#3b87cd}{-2}\vert\times\vert\color{#3a9a38}{x-5}\vert-6\vert x-5\vert\\&=2\color{#fa7921}{\vert x-5\vert}-6\color{#fa7921}{\vert x-5\vert}\\&=-4\vert x-5\vert\end{align}||
Example 3
Again, we apply the property of the absolute value of a product. Then, we perform the division by transforming it into a multiplication. Finally, we simplify the common factors.
||\begin{align}\dfrac{\vert\! -5x+25\vert}{\vert 6x-21\vert}\div\dfrac{\vert x-5\vert}{\vert 2x-7\vert} &=\dfrac{\vert\! -5(x-5)\vert}{\vert 3(2x-7)\vert}\times\dfrac{\vert 2x-7\vert}{\vert x-5\vert}\\ &=\dfrac{\vert\! -5\vert\times\cancel{\vert x-5\vert}\times\cancel{\vert 2x-7\vert}}{\vert3\vert\times\cancel{\vert 2x-7\vert}\times\cancel{\vert x-5\vert}}\\ &=\dfrac{5}{3} \end{align}||
Example 4
In this last example, we again apply the property of the absolute value of a product, and then that of the absolute value of a quotient. Lastly, we simplify the common factors.
||\begin{align}\dfrac{\vert 12 -3x\vert}{\vert 5x-20\vert}&=\dfrac{\vert -3x+12\vert}{\vert5x-20\vert}\\ &=\dfrac{\boldsymbol\vert -3(x-4)\boldsymbol\vert}{\boldsymbol\vert5(x-4)\boldsymbol\vert}\\ &=\left\vert\dfrac{ -3\cancel{(x-4)}}{5\cancel{(x-4)}}\right\vert\\ &=\left\vert\dfrac{ -3}{5}\right\vert\\ &=\dfrac{3}{5} \end{align}||