Rational Numbers & Equations Lesson 4.4 Rational Numbers & Equations The Distributive Property

Warm-Up Solve each equation. Check your solution. 5x – 7 = –17 x = –2 + 10 = 14 –d = –12 x = –2 y = 16 d = 12

The Distributive Property Lesson 4.4 The Distributive Property Use the Distributive Property to simplify expressions. Solve equations using the Distributive Property.

Vocabulary Term A number or the product of a number and a variable. Constant A term with no variable. Coefficient The number multiplied by a variable.

The Distributive Property For any numbers a, b and c: a(b + c) = a(b) + a(c) or ab + ac a(b – c) = a(b) – a(c) or ab – ac

Example 1 Use the Distributive Property to simplify each expression. a. 7(x + 8) = 7(x) + 7(8) 7x + 56 b. –3(y – 2) = –3(y) – (–3)(2) –3y –(–6) –3y + 6 Drawing arrows from the front coefficient to each term inside the parentheses will help guide you.

Multiply the two coefficients together. Think of it as Example 1 Continued… Use the Distributive Property to simplify each expression. c. = Multiply the two coefficients together. Think of it as

Solving Equations Using the Distributive Property Distribute the front coefficient to remove the parentheses. Undo addition or subtraction using inverse operations. Undo multiplication or division using inverse operations. Check your answer.

Example 2 Solve the equation for the variable. Check your solution. a. 3(x + 5) = 33 Write the equation. Distribute 3 through the parentheses. Subtract 15 from each side of the equation. Divide both sides by 3.  Check the solution. 3(x + 5) = 33 3x + 15 = 33 –15 –15 3x = 18 3 3 x = 6 3(6 + 5) 33 3(11) 33 33 = 33

Example 2 Continued… Solve the equation for the variable. Check your solution. b. –5(2m – 1) = 25 Write the equation. Distribution (–5) through the parentheses. Subtract 5 from each side of the equation. Divide both sides by –10.  Check the solution. –5(2m – 1) = 25 –10m + 5 = 25 – 5 –5 –10m = 20 –10 –10 m = –2 –5(2(–2) – 1) ≟ 25 –5(–4 – 1) ≟ 25 –5(–5) ≟ 25 25 = 25

Example 3 Tiffany works as a travel agent. All the employees at her work are required to sell a certain number of vacation packages each day. Tiffany has sold 3 more than the required amount each day for the last 12 days. If she has sold 84 vacation packages, what is the required amount of packages that each employee must sell daily? Let v represent the required number of vacation packages sold daily. Write the equation. Distribute 12 through the parentheses. Subtract 36 from both sides. Divide both sides by 12. Check the solution. Each employee must sell 4 vacation packages daily. 12(v + 3) = 84 12v + 36 = 84 –36 –36 12v = 48 12 12 v = 4 12(4 + 3) = 84 12(7) = 84 84 = 84

Communication Prompt What steps would you take to solve the equation 4(x + 3) = – 8?

Exit Problems Use the Distributive Property to simplify each expression. 1. 3(x – 5) 2. –2(6x + 1) Solve each equation. Show your work and check your solution. 3. 8(x –1) = 16 4. –20 = (6x + 2) 3x – 15 –12x – 2 x = 3 x = –7