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The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.

Simplify: −5 + 3 ( −2 + 7 ) .

Solution

−5 + 3 ( −2 + 7 )
Simplify inside the parentheses. −5 + 3 ( 5 )
Multiply. −5 + 15
Add left to right. 10
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Simplify the expression:

−2 + 5 ( −4 + 7 )

13

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Simplify the expression:

−4 + 2 ( −3 + 5 )

0

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Evaluate variable expressions with integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions    .

Evaluate x + 7 when

  1. x = −2
  2. x = −11 .

Solution

Evaluate x + 7 when x = −2
.
. .
Simplify. .
Evaluate x + 7 when x = −11
.
. .
Simplify. .

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Evaluate each expression for the given values:

x + 5 when

  1. x = −3 and
  2. x = −17

  1. ⓐ 2
  2. ⓑ −12

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Evaluate each expression for the given values: y + 7 when

  1. y = −5
  2. y = −8

  1. ⓐ 2
  2. ⓑ −1

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When n = −5 , evaluate

  1. n + 1
  2. n + 1 .

Solution

Evaluate n + 1 when n = −5
.
. .
Simplify. .
Evaluate n + 1 when n = −5
.
. .
Simplify. .
Add. .

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When n = −8 , evaluate

  1. n + 2
  2. n + 2

  1. ⓐ −6
  2. ⓑ 10

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When y = −9 , evaluate

  1. y + 8
  2. y + 8 .

  1. ⓐ −1
  2. ⓑ 17

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Next we'll evaluate an expression with two variables.

Evaluate 3 a + b when a = 12 and b = −30 .

Solution

.
. .
Multiply. .
Add. .

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Evaluate the expression:

a + 2 b when a = −19 and b = 14 .

9

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Evaluate the expression:

5 p + q when p = 4 and q = −7 .

13

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Evaluate ( x + y ) 2 when x = −18 and y = 24 .

Solution

This expression has two variables. Substitute −18 for x and 24 for y .

( x + y ) 2
. ( −18 + 24 ) 2
Add inside the parentheses. ( 6 ) 2
Simplify 36

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Evaluate:

( x + y ) 2 when x = −15 and y = 29 .

196

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Evaluate:

( x + y ) 3 when x = −8 and y = 10 .

8

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Translate word phrases to algebraic expressions

All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase the sum indicates addition.

Translate and simplify: the sum of −9 and 5 .

Solution

The sum of −9 and 5 indicates addition. the sum of −9 and 5
Translate. −9 + 5
Simplify. −4
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Translate and simplify the expression:

the sum of −7 and 4

−7 + 4 = −3

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Translate and simplify the expression:

the sum of −8 and −6

−8 + (−6) = −14

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Translate and simplify: the sum of 8 and −12 , increased by 3 .

Solution

The phrase increased by indicates addition.

The sum of 8 and −12 , increased by 3
Translate. [ 8 + ( −12 ) ] + 3
Simplify. −4 + 3
Add. −1
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Translate and simplify:

the sum of 9 and −16 , increased by 4 .

[9 + (−16)] + 4 = −3

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Translate and simplify:

the sum of −8 and −12 , increased by 7 .

[−8 + (−12)] + 7 = −13

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Add integers in applications

Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of $5 could be represented as −$5. Let’s practice translating and solving a few applications.

Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.

The temperature in Buffalo, NY, one morning started at 7 degrees below zero Fahrenheit. By noon, it had warmed up 12 degrees . What was the temperature at noon?

Solution

We are asked to find the temperature at noon.

Write a phrase for the temperature. The temperature warmed up 12 degrees from 7 degrees below zero.
Translate to math notation. −7+12
Simplify. 5
Write a sentence to answer the question. The temperature at noon was 5 degrees Fahrenheit.
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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