2.8: Expressions and the Distributive Property
Suppose a piece of giftwrapping paper had a length of 5 feet and a width of \begin{align*}x\end{align*}
Guidance
At the end of the school year, an elementary school teacher makes a little gift bag for each of his students. Each bag contains one class photograph, two party favors, and five pieces of candy. The teacher will distribute the bags among his 28 students. How many of each item does the teacher need?
Example A
You could begin to solve this problem by deciding your variables.
Let \begin{align*}p=photograph, \ f=favors,\end{align*}
Next you can write an expression to represent the situation: \begin{align*}p + 2f + 5c.\end{align*}
There are 28 students in class, so the teacher needs to repeat the bag 28 times. An easier way to write this is \begin{align*}28 \cdot (p + 2f + 5c).\end{align*}
We can omit the multiplication symbol and write \begin{align*}28(p + 2f + 5c)\end{align*}
Therefore, the teacher needs \begin{align*}28p + 28(2f) + 28(5c)\end{align*}
The teacher needs 28 photographs, 56 favors, and 140 pieces of candy to complete the endofyear gift bags.
When you multiply an algebraic expression by another expression, you apply the Distributive Property.
The Distributive Property: For any real numbers \begin{align*}M, \ N,\end{align*}
\begin{align*}&M(N+K)= MN+MK\\
&M(NK)= MNMK\end{align*}
Example B
Determine the value of \begin{align*}11(2 + 6)\end{align*}
Solution: Using the Order of Operations: \begin{align*}11(2 + 6) = 11(8)= 88.\end{align*}
Using the Distributive Property: \begin{align*}11(2 + 6) = 11(2) + 11(6)= 22 + 66 = 88.\end{align*}
Regardless of the method, the answer is the same.
Example C
Simplify \begin{align*}7(3x  5).\end{align*}
Solution 1: Think of this expression as seven groups of \begin{align*}(3x 5)\end{align*}
\begin{align*}(3x5)+(3x5)+(3x5)+(3x5)+(3x5)+(3x5)+(3x5)=21x35\end{align*}
Solution 2: Apply the Distributive Property.
\begin{align*}7(3x5)= 7(3x)+7(5)= 21x35\end{align*}
Guided Practice
Simplify \begin{align*}\frac{2}{7} (3y^2  11).\end{align*}
Solution: Apply the Distributive Property.
\begin{align*}&\frac{2}{7} (3y^2 + 11)= \frac{2}{7} (3y^2) + \frac{2}{7}(11)=\\
&\frac{2}{7} \frac{(3y^2)}{1} + \frac{2}{7}\frac{(11)}{1}=\frac{2 \times 3y^2}{7 \times 1} + \frac{2\times (11)}{7\times 1}=\\
&\frac{6y^2}{7}+\frac{22}{7}=\frac{6y^2}{7}\frac{22}{7}\end{align*}
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Distributive Property (5:39)
Use the Distributive Property to simplify the following expressions.

\begin{align*}(x + 4)  2(x + 5)\end{align*}
(x+4)−2(x+5) 
\begin{align*}\frac{1}{2}(4z + 6)\end{align*}
12(4z+6) 
\begin{align*}(4 + 5)(5 + 2)\end{align*}
(4+5)−(5+2)  \begin{align*}(x + 2 + 7)\end{align*}
 \begin{align*}0.25 (6q + 32)\end{align*}
 \begin{align*}y(x + 7)\end{align*}
 \begin{align*}4.2(h  11)\end{align*}
 \begin{align*}13x(3y + z)\end{align*}
 \begin{align*}\frac{1}{2}(x  y)  4\end{align*}
 \begin{align*}0.6(0.2x + 0.7)\end{align*}
 \begin{align*}(2  j)(6)\end{align*}
 \begin{align*}4(m + 7) 6(4  m)\end{align*}
 \begin{align*}5(y  11) + 2y\end{align*}
Mixed Review
 Translate the following into an inequality: Jacob wants to go to Chicago for his class trip. He needs at least $244 for the bus, hotel stay, and spending money. He already has $104. How much more does he need to pay for his trip?
 Underline the math verb(s) in this sentence: 6 times a number is 4 less than 16.
 Draw a picture to represent \begin{align*}3 \frac{3}{4}\end{align*}.
 Determine the change in \begin{align*}y\end{align*} in the equation \begin{align*}y = \frac{1}{6} x4\end{align*} between \begin{align*}x=3\end{align*} and \begin{align*}x=9\end{align*}.
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Image Attributions
Here you'll learn to simplify an expression in the form @$\begin{align*}M(N+K)\end{align*}@$ or @$\begin{align*}M(NK)\end{align*}@$ by using the Distributive Property.