Evaluating Expressions Using the Distributive Property

Learning Outcomes

  • Evaluate expressions using the distributive property
  • Evaluate expressions containing absolute value

Using the Distributive Property With the Order of Operations

Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.


Simplify: [latex]8 - 2\left(x+3\right)[/latex]


Distribute. [latex]8-2\cdot x-2\cdot 3[/latex]
Multiply. [latex]8-2x-6[/latex]
Combine like terms. [latex]-2x+2[/latex]


try it



Simplify: [latex]4\left(x - 8\right)-\left(x+3\right)[/latex]


try it

In the following example, we simplify more expressions that require the distributive property.

Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part 1, we will evaluate the form with parentheses, and in part 2 we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.


When [latex]y=10[/latex] evaluate:
1. [latex]6\left(5y+1\right)[/latex]
2. [latex]6\cdot 5y+6\cdot 1[/latex]


try it



When [latex]y=3[/latex], evaluate
1. [latex]-2\left(4y+1\right)[/latex]
2. [latex]-2\cdot 4y+\left(-2\right)\cdot 1[/latex]


try it



When [latex]y=35[/latex] evaluate
1. [latex]-\left(y+5\right)[/latex]

2. [latex]-y-5[/latex] to show that [latex]-\left(y+5\right)=-y-5[/latex]


try it

The following video provides another way to show that the distributive property works.

Absolute Value

Absolute value expressions are another method of grouping that you may see. Recall that the absolute value of a quantity is always positive or [latex]0[/latex].

When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done.


Simplify [latex]\dfrac{3+\left|2-6\right|}{2\left|3\cdot1.5\right|-\left(-3\right)}[/latex]

Try It

The following video uses the order of operations to simplify an expression in fraction form that contains absolute value terms. Note how the absolute values are treated like parentheses and brackets when using the order of operations.


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