Algebra 1 unit 1.2

UNIT 1.2 ORDER OF OPERATIONSUNIT 1.2 ORDER OF OPERATIONS

Warm UpSimplify.

1. 42 2. |5 – 16| 3. –23 4. |3 – 7|16 –8 4

Translate each word phrase into a numerical or algebraic expression.

5. the product of 8 and 6

6. the difference of 10y and 4

8 × 6

10y – 4

11

Simplify each fraction.

7. 8.16 2

8568

17

Use the order of operations to simplify expressions.

Objective

order of operations

Vocabulary

When a numerical or algebraic expression containsmore than one operation symbol, the order of operations tells which operation to perform first.

First:

Second:

Third:

Fourth:

Perform operations inside grouping symbols.

Evaluate powers.

Perform multiplication and division from left to right.

Perform addition and subtraction from left to right.

Order of Operations

Grouping symbols include parentheses ( ), brackets [ ], and braces { }. If an expression contains more than one set of grouping symbols, evaluate the expression from the innermost set first.

Helpful Hint The first letter of these words can help you remember the order of operations.

Please

Excuse

My

Dear

Aunt

Sally

Parentheses

Exponents

Multiply

Divide

Add

Subtract

Simplify each expression.A. 15 – 2 · 3 + 1

15 – 2 · 3 + 115 – 6 + 1

10

There are no grouping symbols.

Multiply.

Subtract and add from left to right.

B. 12 – 32 + 10 ÷ 212 – 32 + 10 ÷ 2

12 – 9 + 10 ÷ 2

12 – 9 + 5

8

There are no grouping symbols.

Evaluate powers. The exponent applies only to the 3.

Divide.Subtract and add from left to right.

Example 1: Translating from Algebra to Words

8 ÷ · 3

There are no groupingsymbols.

Divide.

48 Multiply.

Check It Out! Example 1a

Simplify the expression.1 2

8 ÷ · 3 1 2

16 · 3

5.4 – 32 + 6.2

5.4 – 32 + 6.2 There are no groupingsymbols.

5.4 – 9 + 6.2 Simplify powers.

–3.6 + 6.2

2.6

Subtract

Add.

Check It Out! Example 1b

Simplify the expression.

–20 ÷ [–2(4 + 1)]

–20 ÷ [–2(4 + 1)] There are two sets of groupingsymbols.

–20 ÷ [–2(5)] Perform the operations in theinnermost set.

–20 ÷ –10

2

Perform the operation insidethe brackets.

Divide.

Check It Out! Example 1c

Simplify the expression.

Example 2A: Evaluating Algebraic Expressions

Evaluate the expression for the given value of x.

10 – x · 6 for x = 3

10 – x · 6

10 – 3 · 6

10 – 18

–8

First substitute 3 for x.

Multiply.Subtract.


42(x + 3) for x = –2

42(x + 3)

42(–2 + 3)

42(1)

16(1)

16

First substitute –2 for x.

Perform the operation inside the parentheses.

Evaluate powers.

Multiply.

Example 2B: Evaluating Algebraic Expressions


14 + x2 ÷ 4 for x = 2

Check It Out! Example 2a

14 + x2 ÷ 4

14 + 22 ÷ 4

14 + 4 ÷ 4

14 + 1


Square 2.

Divide.

Add.15

(x · 22) ÷ (2 + 6) for x = 6

Check It Out! Example 2b


(x · 22) ÷ (2 + 6)

(6 · 22) ÷ (2 + 6)

(6 · 4) ÷ (2 + 6)

(24) ÷ (8)

3


Square two.

Perform the operations inside the parentheses.

Divide.

Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division.

Simplify.2(–4) + 22 42 – 9

2(–4) + 22 42 – 9 –8 + 22 42 – 9

–8 + 22 16 – 9

14 7

2

Example 3A: Simplifying Expressions with Other Grouping Symbols

The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing.

Multiply to simplify the numerator.

Evaluate the power in the denominator.

Add to simplify the numerator. Subtract to simplify the denominator.

Divide.

Example 3B: Simplifying Expressions with Other Grouping Symbols

Simplify.

3|42 + 8 ÷ 2|

3|42 + 8 ÷ 2|

3|16 + 8 ÷ 2|

3|16 + 4|

3|20|

3 · 20

60

The absolute-value symbols act as grouping symbols.

Evaluate the power.Divide within the absolute-value symbols.

Add within the absolute-symbols.

Write the absolute value of 20.

Multiply.

Check It Out! Example 3aSimplify.5 + 2(–8)

(–2) – 3 3

5 + 2(–8)

–8 – 3

5 + 2(–8)

(–2) – 3 3

5 + (–16)

– 8 – 3

–11–11

1

The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing.

Evaluate the power in the denominator.

Multiply to simplify the numerator.

Add.

Divide.

Check It Out! Example 3bSimplify.

|4 – 7|2 ÷ –3

|4 – 7|2 ÷ –3

|–3|2 ÷ –3

32 ÷ –3

9 ÷ –3

–3

The absolute-value symbols act as grouping symbols.

Subtract within the absolute-value symbols.

Write the absolute value of –3.

Divide.

Square 3.

Check It Out! Example 3c

Simplify.

The radical symbol acts as a grouping symbol.

Subtract.

Take the square root of 49.

Multiply.

3 · 7

21

You may need grouping symbols when translating from words to numerical expressions.

Remember!

Look for words that imply mathematical operations.

difference subtractsum addproduct multiplyquotient divide

Example 4: Translating from Words to Math


A. the sum of the quotient of 12 and –3 and the square root of 25

Show the quotient being added to .

B. the difference of y and the product of 4

and

Use parentheses so that the product is evaluated first.

Check It Out! Example 4

Translate the word phrase into a numerical or algebraic expression: the product of 6.2 and the sum of 9.4 and 8.

6.2(9.4 + 8)Use parentheses to show that the sum of 9.4 and 8 is evaluated first.

Example 5: Retail Application

A shop offers gift-wrapping services at three price levels. The amount of money collected for wrapping gifts on a given day can be found by using the expression 2B + 4S + 7D. On Friday the shop wrapped 10 Basic packages B, 6 Super packages S, and 5 Deluxe packages D. Use the expression to find the amount of money collected for gift wrapping on Friday.

Example 5 Continued

2B + 4S + 7D

2(10) + 4(6) + 7(5)

20 + 24 + 35

44 + 35

79

First substitute the value for each variable.

Multiply.

Add from left to right.

Add.

The shop collected $79 for gift wrapping on Friday.

Check It Out! Example 5

Another formula for a player's total number of bases is Hits + D + 2T + 3H. Use this expression to find Hank Aaron's total bases for 1959, when he had 223 hits, 46 doubles, 7 triples, and 39 home runs.

Hits + D + 2T + 3H = total number of bases

223 + 46 + 14 + 117

223 + 46 + 2(7) + 3(39) First substitute values for each variable.Multiply.

400 Add.

Hank Aaron’s total number of bases for 1959 was 400.

Lesson QuizSimply each expression.

1. 2[5 ÷ (–6 – 4)] 2. 52 – (5 + 4)

|4 – 8|3. 5 × 8 – 4 + 16 ÷ 22

–1 4

40


4. 3 three times the sum of –5 and n 3(–5 + n)

5. the quotient of the difference of 34 and 9 and the square root of 25

6. the volume of a storage box can be found using the expression l · w(w + 2). Find the volume of the box if l = 3 feet and w = 2 feet.

24 cubic feet

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Algebra 1 unit 1.2

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