1.–15 + (–19) + 16 = ANSWER –18 2.6(–x)(–4) = ANSWER 24x Warm-Up for Lesson 2.6 ? ?

3.–9(–2)(–4b) = ANSWER –72b 4. Kristin paid $1.90 per black-and-white photo b and $6.80 per color photo c to have some photos restored. What was the total amount A that she paid if she had 8 black-and-white and 12 color photos restored? ANSWER $96.80 Warm-Up for Lesson 2.6 ?

T HE D ISTRIBUTIVE P ROPERTY a(b + c) = ab + ac (b + c)a = ba + ca 2(x + 5)2(x) + 2(5)2x + 10 (x + 5)2(x)2 + (5)22x + 10 (1 + 5x)2(1)2 + (5x) x y(1 – y)y(1) – y(y)y – y 2 U SING THE D ISTRIBUTIVE P ROPERTY = = = = = = = = The product of a and (b + c):

(y – 5)(–2)= (y)(–2) + (–5)(–2) = –2y + 10 –(7 – 3x)= (–1)(7) + (–1)(–3x) = –7 + 3x = –3 – 3x (–3)(1 + x) = (–3)(1) + (–3)(x) Simplify. Distribute the –3. Simplify. Distribute the –2. Simplify. –a = –1 a U SING THE D ISTRIBUTIVE P ROPERTY Remember that a factor must multiply each term of an expression. Forgetting to distribute the negative sign when multiplying by a negative factor is a common error.

Use the distributive property to write an equivalent expression. EXAMPLE 1 Apply the distributive property a. 4(y + 3) = b. (y + 7)y = d. (2 – n)8 = c. n(n – 9) = 4y + 12 y 2 + 7y n 2 – 9n 16 – 8n

= – 15y + 3y 2 b. (5 – y)(–3y) = Simplify. Distribute – 3y. = – 2x – 14 Distribute – 2. Use the distributive property to write an equivalent expression. EXAMPLE 2 Distribute a negative number a. –2(x + 7)= – 2(x) + – 2(7) 5(–3y) – y(–3y)

Simplify. = (– 1)(2x) – (–1)(11) c. –(2x – 11) = of – 1 Multiplicative property EXAMPLE 2 Distribute a negative number Distribute – 1. = – 2x + 11 (–1)(2x – 11)

GUIDED PRACTICE for Examples 1, 2 and 3 Use the distributive property to write an equivalent expression. 1. 2(x + 3) = 2x – (4 – y) = – 4 + y 3. (m – 5)(– 3m) = – 3m m 4. (2n + 6) 1 2 = n + 3

S IMPLIFYING BY C OMBINING L IKE T ERMS Each of these terms is the product of a number and a variable.terms +–3y2y2 x +–3y2y2 x number +–3y2y2 x variable. +–3y2y2 x –1 is the coefficient of x. 3 is the coefficient of y 2. x is the variable. y is the variable. Each of these terms is the product of a number and a variable. x2x2 x2x2 y3y3 y3y3 Like terms have the same variable raised to the same power. y 2 – x 2 + 3y 3 – – 3x 2 + 4y 3 + y variablepower.Like terms The constant terms –5 and 3 are also like terms.

Constant terms: – 4, 2 Coefficients: 3, – 6 Like terms: 3x and – 6x; – 4 and 2 Identify the terms, like terms, coefficients, and constant terms of the expression 3x – 4 – 6x + 2. Write the expression as a sum: 3x + (–4) + (–6x) + 2 SOLUTION EXAMPLE 3 Identify parts of an expression Terms: 3x, – 4, – 6x, 2

GUIDED PRACTICE for Examples 1, 2 and 3 Identify the terms, like terms, coefficients, and constant terms of the expression – 7y + 8 – 6y – 13. Coefficients: – 7, – 6 Like terms: – 7y and – 6y, 8 and – 13; Constant terms: 8, – 13 Terms: – 7y, 8, – 6y, – 13 ANSWER

Combine like terms. S IMPLIFYING BY C OMBINING L IKE T ERMS 4x – x 2 = (8 + 3)x Use the distributive property. = 11x Add coefficients. 8x + 3x = Group like terms. Rewrite as addition expression. Distribute the –2. Multiply. Combine like terms and simplify. 4x 2 – x = 3x – 2(4 + x) = 3 + (–2)(4 + x) = 3 + [(–2)(4) + (–2)(x)] = 3 + (–8) + (–2x) = –5 + (–2x) = –5 – 2x

Standardized Test Practice EXAMPLE 4 ANSWER The correct answer is B. DCBA Simplify the expression 4(n + 9) – 3(2 + n). 4(n + 9) – 3(2 + n) = Distributive property = n + 30 Combine like terms. A B C D n + 3 5n + 30 n + 305n + 3 4n + 36 – 6 – 3n

Solve a multi-step problem EXAMPLE 5 EXERCISING Your daily workout plan involves a total of 50 minutes of running and swimming. You burn 15 calories per minute when running and 9 calories per minute when swimming. Let r be the number of minutes that you run. Find the number of calories you burn if you run for 20 minutes. SOLUTION The workout lasts 50 minutes, and your running time is r minutes. So, your swimming time is (50 – r) minutes.

Solve a multi-step problem EXAMPLE 5 STEP 1 C = Write equation. = 15 r – 9r Distributive property = 6r Combine like terms. Write a verbal model. Then write an equation. 15r + 9(50 – r) C = 5 r + 9 (50 – r) Amount burned (calories) Burning rate when running (calories/minute) Running time (minutes) Swimming time (minutes) = + Burning rate when swimming (calories/minute)

Solve a multi-step problem EXAMPLE 5 C = Write equation. = 6(20) = 570 Substitute 20 for r. Then simplify. ANSWER You burn 570 calories in your 50 minute workout if you run for 20 minutes and swim for 30 minutes. STEP 2 Find the value of C when r = 20. 6r + 450

GUIDED PRACTICE for Examples 4 and 5 6. Simplify the expression 5(6 + n) – 2(n – 2) = n. 7. WHAT IF ? In Example 5, suppose your workout lasts 45 minutes. How many calories do you burn if you run for 20 minutes ? 30 minutes ? ANSWER You burn 525 calories in your 45 minute workout if you run for 20 minutes. You burn 585 calories in your 45 minute workout if you run for 30 minutes.