Do you distribute before or after absolute value?
When solving equations or simplifying expressions involving absolute value, it is essential to understand the proper order of operations. One frequently asked question is whether to distribute before or after dealing with absolute value. Before we delve into the answer, let’s first clarify what absolute value is and how it is used.
Absolute value, denoted by two vertical bars (|x|), represents the distance between a number x and zero on the number line. It disregards the positive or negative sign, resulting in a non-negative value. For example, |6| equals 6, while |-6| also equals 6. Now that we have a clear understanding of absolute value, let’s address the question at hand.
Do you distribute before or after absolute value?
The answer is simple: distribute after dealing with absolute value.
When faced with an expression that contains absolute value and multiplication or division, it is crucial to apply the absolute value first. Once that is done, you can then distribute any multiplication or division operation.
To illustrate this, let’s consider the following example:
|-3x + 7| * 2
First, we evaluate the absolute value by removing the outer bars and applying the non-negative property of absolute value. This results in:
(3x – 7) * 2
Now that the absolute value is removed, we can proceed with distributing the multiplication:
6x – 14
The distribution step ensures that every term within the parentheses is multiplied or divided by the factor outside the parentheses.
Now that we’ve answered the primary question, let’s tackle some related FAQs to further clarify the concept:
FAQs about distributing before or after absolute value:
1.
What happens if I distribute before dealing with absolute value?
Distributing before absolute value can lead to incorrect solutions as you may mistakenly apply the multiplication or division operation to negative values.
2.
Is it ever possible to distribute before dealing with absolute value?
In certain cases, where the absolute value expression is entirely contained within parentheses and is preceded by a negative sign, you can distribute and simplify before dealing with absolute value.
3.
What is the purpose of absolute value in the order of operations?
Absolute value ensures that the result is non-negative, regardless of the sign of the expression within it.
4.
Can I distribute addition or subtraction before absolute value?
No, because addition or subtraction cannot be distributed across the absolute value function.
5.
How do I know when I should apply absolute value?
Look for expressions or equations where an absolute value function is present. Evaluate the absolute value first by disregarding the sign and then distribute if necessary.
6.
What if there are multiple absolute value expressions in an equation?
Treat each absolute value separately and apply the order of operations within each individual expression.
7.
Can I simplify an expression with both multiplication and addition before dealing with absolute value?
No, the order of operations dictates that absolute value should be dealt with first, followed by multiplication or division, and then addition or subtraction.
8.
Can I distribute before or after evaluating the expression inside the absolute value?
Distribute operations can only be applied after the absolute value has been evaluated and reduced to its non-negative form.
9.
Is it necessary to distribute if the expression inside the absolute value is already simplified?
If the expression inside the absolute value is already fully simplified, there is no need to distribute any further.
10.
Can I use the distributive property on an equation with an absolute value function?
No, the distributive property applies to multiplying or dividing terms within an expression, not to equations as a whole.
11.
What happens if I distribute incorrectly after applying absolute value?
Incorrect distribution can lead to incorrect solutions or expressions that do not represent the original equation accurately.
12.
Are there any shortcuts or rules of thumb to remember when dealing with absolute value?
The key rule to remember is to always evaluate absolute value first and distribute afterward when dealing with multiplication or division.