Chapter 6—Expressions and Equations 6-5 Removing Parentheses
When you complete the work for this section, you should be able to: - Use the distributive property of multiplication and addition to remove parentheses from equations.
- Simplify equations by removing parentheses and combining like terms.
| When you see an algebraic expression in a science or business textbook, it is usually presented in its simplest form. Compare these two versions of the same expression: 2x + 3(x + 4) and 5x + 12 These expressions are the same. In other words, 2x + 3(x + 4) = 5x + 12. You can prove this fact by substituting any value for the x's and solving the results. You can see that the expression, 5x + 2 is clearly simpler than the version that includes parentheses. The purpose of this lesson is to demonstrate how to simplify expressions by removing the parentheses. This procedure uses the distributive property of multiplication and addition. | Recall The distributive property shows the relationship between the product of one term times the sum of two terms, a(b + c). a(b + c) = ab + ac where a, b, and c are real numbers. | Removing Parentheses from Expressions of Form a(bx + c) Removing the parentheses from an expression of the form a(bx + c) is a matter of applying the basic distributive property: Examples A - 2(y + 6) = 2y + 12
- 1(x + 2) = x + 2
The procedure for removing parentheses require a bit more care when negative values and subtraction are involved. Examples B - 2(y – 6) = 2y – 12
- 4(3 – P) = 12 – 4P
– 3(z + 2) = – 3z – 6 – 4(y – 1) = – 4y + 4 Examples and Exercises | Removing Parentheses Use these interactive examples and exercises to strengthen your understanding and build your skills: | | Removing Sets of Parentheses that are Added/Subtracted Consider this messy looking expression: 2(x + 1) + 3(x + 6) Working from left to right: 2(x + 1) + 3(x + 6) = 2x + 2 + 3x + 18 Collecting like terms: 2x + 2 + 3x + 18 = 5x + 20 Result: 2(x + 1) + 3(y + 6) = 5x + 20 Examples C Problem 1 Remove the parentheses: 2 + 8(a + 3) + 2(3a – 1) | | | Procedure | | - Use the distributive property to remove the parentheses.
| 2 + 8(a + 3) + 2(3a – 1) = 2 + 8a + 24 + 6a – 2 | - Collect like terms.
| 2 + 8a + 24 + 6a – 2 = 14a + 24 | | Solution | | 2 + 8(a + 3) + 2(3a – 1) = 14a + 24 | | Problem 2 Remove the parentheses: 1 + 6(x + 2) – 4(5x + 7) | | | Procedure | | - Use the distributive property to remove the parentheses.
| 1 + 6(x + 2) – 4(5x + 7) = 1 + 6x + 12 – 20x – 28 | - Collect like terms.
| 1 + 6x + 12 – 20x – 28 = –14x – 15 | | Solution | | 1 + 6(x + 2) – 4(5x + 7) = –14x – 15 | | Problem 3 Remove the parentheses: 2y – 4(y – 1) + 4(5y + 3) | | | Procedure | | - Use the distributive property to remove the parentheses.
| 2y – 4(y – 1) + 4(5y + 3) = 2y – 4y + 4 + 20y + 12 | - Collect like terms.
| 2y – 4y + 4 + 20y + 12 = 18y + 16 | | Solution | | 2y – 4(y – 1) + 4(5y + 3) = 18y + 16 | | Problem 4 Remove the parentheses: y + 2(x – 5) + (5y + 3x) | | | Procedure | | - Use the distributive property to remove the parentheses.
| y + 2(x – 5) + (5y + 3x) = y + 2x – 10 + 5y + 3x | - Collect like terms.
| 5x + 6y – 10 | | Solution | | y + 2(x – 5) + (5y + 3x) = 5x + 6y – 10 | | Examples and Exercises | Removing Parentheses Use these interactive examples and exercises to strengthen your understanding and build your skills: | | Removing Sets of Parentheses that are Multiplied  (a + b)(c + d) a(c + d) + b(c + d) ac + ad + bc + bd (2 + 4)(3 + 5) 2(3 + 5) + 4(3 + 5) 6 + 10 + 12 + 20 = 48 (2 + 4)(3 + 5) = 48 Note: This is the same as (2 + 4)(3 + 5) = (6)(8) = 48 (3 – 6)(2 + 5) 3(2 + 5) – 6(2 + 5) 6 + 15 – 12 – 30 = (3 – 6)(2 + 5) = – 21 (2x + 1)(4x – 1) 8x2 – 2x + 4x – 1 8x2 + 2x – 1
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