Section 1.4

Subtraction of Real Numbers

Objective: Subtract positive and negative real numbers.

1.4 Lecture Guide: Subtraction of Real Numbers

Phrases Used To Indicate Subtraction:Key Phrase Verbal Example Algebraic Example

Minus "x minus y"

Difference "The difference between 12 and 8"

Decreased by "An interest rate r is decreased by 0.5%"

Less than "7 less than x"

Change "The change from

7x

0.005r

12 8

x y

9670to

96 70

1.

Translate each verbal statement into algebraic form.

t minus seven

2.

Translate each verbal statement into algebraic form.

is decreased by three x

3.

Translate each verbal statement into algebraic form.

Four less than y

It is helpful when first performing a subtraction to actually rewrite the subtraction as addition. This is not generally done once you are comfortable with performing subtraction mentally.

Subtraction

Verbally

Numerical Example

AlgebraicallyFor any real numbers x and y,

6 4 6 ______

______

To subtract y from x, add the opposite of y to x.

x y x y

Algebraically

Verbally

Numerical Example

Subtracting Fractions

To subtract fractions with the same denominator, subtract the numerators and use the common denominator.

a c a c

b b b

0bfor

6 2

7 7

Algebraically

Verbally

for

To subtract fractions with different denominators, first express each fraction in terms of a common denominator and then subtract the numerators using this common denominator

a c ad bc ad bc

b d bd bd bd

0b and 0d

Numerical Example 3 1

4 3

Subtracting Fractions

4.

The terms in each expression have the same sign. First rewrite each difference as a sum and then give its value.

8 1

Example

12 5

12 5

7

Example

7 15

7 15

8

5.

The terms in each expression have the same sign. First rewrite each difference as a sum and then give its value.

13 9

6.

The terms in each expression have the same sign. First rewrite each difference as a sum and then give its value.

17 17

7.

The terms in each expression have the same sign. First rewrite each difference as a sum and then give its value.

7.3 5.1

8.

The terms in each expression have the same sign. First rewrite each difference as a sum and then give its value.

0.7 3.4

9.

The terms in each expression have the same sign. First rewrite each difference as a sum and then give its value.

4 4

9 9

10.

The terms in each expression have the same sign. First rewrite each difference as a sum and then give its value.

5 1

9 3

11.

The terms in each expression have the same sign. First rewrite each difference as a sum and then give its value.

3 3

5 8

12. What is the difference between the addition and subtraction symbols and positive and negative signs?

is the change of sign key and

is the subtraction key.

Don’t forget that on a TI-84 Plus calculator

See Calculator Perspective 1.4.2.

13.

The terms in each expression have opposite signs. First rewrite each difference as a sum and then give its value.

8 1

Example Example 7 15

7 15

22

12 5

12 5

17

14.

The terms in each expression have opposite signs. First rewrite each difference as a sum and then give its value.

13 9

15.

The terms in each expression have opposite signs. First rewrite each difference as a sum and then give its value.

17 17

16.

The terms in each expression have opposite signs. First rewrite each difference as a sum and then give its value.

7.3 5.1

17.

The terms in each expression have opposite signs. First rewrite each difference as a sum and then give its value.

0.7 3.4

18.

The terms in each expression have opposite signs. First rewrite each difference as a sum and then give its value.

4 4

9 9

19.

The terms in each expression have opposite signs. First rewrite each difference as a sum and then give its value.

5 1

9 3

20.

The terms in each expression have opposite signs. First rewrite each difference as a sum and then give its value.

3 3

5 8

21. Using both positive and negative 5 and 8, how many different values can you create by adding or subtracting these numbers? Give a written description to generalize what you have found.

22. Determine the value of each expression.

(a) (b)

(c) (d)

7 2 2 7

7 2 2 7

23. We know that addition is commutative. What about subtraction?

24. The number of riders on the Top Thrill Dragster at Cedar Point amusement park is shown in the chart below. Determine the change in the number of riders from 2003 to 2005.

Riders on the Top Thrill Dragster

953,945943,313

562,438

0

500,000

1,000,000

1,500,000

2003 2004 2005

Objective: Calculate the terms of a sequence. A sequence is an ordered set of numbers with a first number, a second number, a third number, etc. Subscript notation often is used to denote the terms of a sequence: 1a , 2a , and na . These terms are read a sub one, a sub two, and a sub n, respectively. If a sequence follows a predictable pattern, then we may be able to describe this pattern with a formula for na . Consider the sequence 5, 4, 3, 2. Here, 1 5a , 2 4a ,

3 3a , and 4 2a .

25. 5na n

Use each formula to calculate the first five terms

1a , 2a , and 5a . , 3a , 4a

26.

Use each formula to calculate the first five terms

1a , 2a , and 5a . , 3a , 4a

3na n

Objective: Check a possible solution of an equation. A solution of an equation is a value for the variable that satisfies the equation. This means that when the value is substituted for the variable, the expressions on each side of the equation will have the ____________ value

Check both and to determine whether either is a solution of the following equations. Then use your calculator to check your results. See Calculator Perspective 1.4.2 for help.

3x 5x

3 8x 27.

(a) Check 3x (b) Check 5x

Check both and to determine whether either is a solution of the following equations. Then use your calculator to check your results. See Calculator Perspective 1.4.2 for help.

3x 5x

28.

(a) Check 3x (b) Check 5x

1 2x x x