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# 2.8: Expressions and the Distributive Property

Difficulty Level: At Grade Created by: CK-12
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Practice Expressions and the Distributive Property
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Suppose a piece of gift-wrapping paper had a length of 5 feet and a width of \begin{align*}x\end{align*} feet. When wrapping a present, you cut off a vertical strip of 1 foot so that the width of the piece of gift-wrapping paper decreased by 1 foot. What would be the area of the remaining paper in square feet? Could you write this expression in simplified form? In this Concept, you'll learn to use the Distributive property so that you can simplify expressions like the one representing this area.

### 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*} and \begin{align*}c=candy.\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*} or \begin{align*}28p + 56f + 140c.\end{align*}

The teacher needs 28 photographs, 56 favors, and 140 pieces of candy to complete the end-of-year 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*} and \begin{align*}K\end{align*}:

\begin{align*}&M(N+K)= MN+MK\\ &M(N-K)= MN-MK\end{align*}

#### Example B

Determine the value of \begin{align*}11(2 + 6)\end{align*} using both Order of Operations and the Distributive Property.

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*}. You could write this expression seven times and add all the like terms.

\begin{align*}(3x-5)+(3x-5)+(3x-5)+(3x-5)+(3x-5)+(3x-5)+(3x-5)=21x-35\end{align*}

Solution 2: Apply the Distributive Property.

\begin{align*}7(3x-5)= 7(3x)+7(-5)= 21x-35\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. CK-12 Basic Algebra: Distributive Property (5:39)

Use the Distributive Property to simplify the following expressions.

1. \begin{align*}(x + 4) - 2(x + 5)\end{align*}
2. \begin{align*}\frac{1}{2}(4z + 6)\end{align*}
3. \begin{align*}(4 + 5)-(5 + 2)\end{align*}
4. \begin{align*}(x + 2 + 7)\end{align*}
5. \begin{align*}0.25 (6q + 32)\end{align*}
6. \begin{align*}y(x + 7)\end{align*}
7. \begin{align*}-4.2(h - 11)\end{align*}
8. \begin{align*}13x(3y + z)\end{align*}
9. \begin{align*}\frac{1}{2}(x - y) - 4\end{align*}
10. \begin{align*}0.6(0.2x + 0.7)\end{align*}
11. \begin{align*}(2 - j)(-6)\end{align*}
12. \begin{align*}4(m + 7) -6(4 - m)\end{align*}
13. \begin{align*}-5(y - 11) + 2y\end{align*}

Mixed Review

1. 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?
2. Underline the math verb(s) in this sentence: 6 times a number is 4 less than 16.
3. Draw a picture to represent \begin{align*}3 \frac{3}{4}\end{align*}.
4. Determine the change in \begin{align*}y\end{align*} in the equation \begin{align*}y = \frac{1}{6} x-4\end{align*} between \begin{align*}x=3\end{align*} and \begin{align*}x=9\end{align*}.

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### Vocabulary Language: English Spanish

Distributive Property

For any real numbers $M, \ N,$ and $K$: $&M(N+K)= MN+MK\\ &M(N-K)= MN-MK$

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