Chapter 01 – Section 05

Open Sentences

© William James Calhoun

To solve open sentences by performing arithmetic operations.

• open sentence - a mathematical statement with one or more variables– the variable has to be replaced by a value

• solving the open sentence - the process of finding out what number the variable is holding place for

• solution - the replacement value for the variable in an open sentence

CLARIFICATION: An open sentence in mathematics is like an English sentence with a blank in it. The sentence:

__________________ is a color.

is true if Blue is placed in the blank, false if Hard is placed in the blank, and neither true nor false if the blank is left empty.

© William James Calhoun

• replacement set - a set of numbers (one or more) which make an open sentence true

• set - a collection of objects or numbers

• element - each individual object in a set

• solution set - all the replacement values that make an open sentence become true

More terms you will be using in this section:

© William James Calhoun

The solution set is the group of numbers that “worked” when they were plugged in.

Solution sets are the values that make the sentence a true statement.

The numbers 4, 5, 6, 7, and 8 are all elements of this set:

{4, 5, 6, 7, 8}

Notice that we use brackets { } to surround sets when we list them.

A replacement set is the group of numbers you will be plugging in to look for the solutions.

© William James Calhoun

EXAMPLE 1α: Find the solution set for y + 5 < 7 if the replacement set is {0, 1, 2, 3, 4}.

EX1EX1β

The solution set for y + 5 7 is {0, 1, 2}.

For these problems, it will help to make a chart.

Replacement set: y + 5 7? True of False? 0 7?

1 7? 2 7? 3 7? 4 7?

0 + 5 = 51 + 5 = 62 + 5 = 73 + 5 = 84 + 5 = 9

TRUETRUETRUEFALSEFALSE

Word of Caution: The book uses “0” to answer solutions sets of {}. To say 0 is the solution set is not the same as saying {0} is the solution set. Plain 0 means there were no true replacements. {0} means the only true replacement was 0 from the replacement set.

© William James Calhoun

EXAMPLE 1β: Find the solution set for if the replacement set is {0, 2, 4, 6, 8}.

Replacement set: (x + 3)/4 >3? True of False? 0 >3?

2 >3? 4 >3? 6 >3? 8 >3?

x 33

4

© William James Calhoun

EXAMPLE 2β: Solve .d223)53(5

We can use order of operations to find values for variables when we set a single variable equal to algebraic expressions.

d = 5or

The solution is 5.

Focus on the numerator.

5(8)

Multiply.

Focus on the denominator.

5(3 + 5)Parenthesis first.

40

6 + 2

Add.

3 · 2 + 2Multiply first.

8

Numerator divided by denominator.

408

KEY NOTE: You can only solve for one variable with one equation.

EX2EX2β

© William James Calhoun

EXAMPLE 2β: Solve each equation.

a. b.7 7 2

m10 7

4.2 2(3.9)

k3.1 2.9

© William James Calhoun

PAGE 35#13 – 31 odd