Transcript Document

Open Sentences
Vocabulary
• Open Sentence – a mathematical statement
(sentence) that contains one or more variables, or
unknown numbers
• An open sentence is neither true nor false until the
variable(s) have been replaced by intended values
called replacement sets
• Replacement Set – intended values or a set of
numbers that are substituted into an equation to
determine if they are solutions or if they satisfy the
open sentence
Vocabulary
• Solving the open sentence means finding a value from the
replacement set that will make the open sentence a true
statement.
• A solution means the replacement number that actually
makes the equation a true statement.
• An open sentence may have one solution, more than one
solution, or no solutions.
Vocabulary
• Set means a collection of objects or numbers and are
represented by using braces {}.
• Element means each object or number in the set is called an
element, or member of the set.
• Sets are named by using capital letters. Examples of sets are
A = {1,2,3,}; B = {6,8,10}; C= {1,2,3,6,8,10}.
• The Solution set of an open sentence is the set of all
replacements for the variable that will satisfy or make the
equation true.
Vocabulary: Equation
• Equation – an equation states that two expressions are
equal. The expressions can be variable or numeric and are
represented on each side of an equal sign.
Equations are
separated by
an equal sign
Variable
Expression
n  310
Numeric
Expression
Open Sentence Example 1
State whether the equation is true or false for the given value
of the variable.
1. 15  x  18; x  3
15  3  18
Substitute 3 into the equation
for the variable x and solve.
True when x = 3
2. 15  x  18;x  7
15 7 18
False when x = 7
Substitute 7 into the equation
for the variable x and solve.
Separating an
equation by a
semi-colon and
indicating the
value of the
variable means
to substitute the
number into the
equation to see
if it is a true
solution.
Open Sentence Example 2
State whether the equation is true or false for the given value of
the variable.
1. 6 x  18; x  3
16  3  18
Substitute 3 into the equation
for the variable x and solve.
True when x = 3
2. 32  x = 15; x  2
32 2 15
False when x = 2
Substitute 2 into the equation
for the variable x and solve.
Separating an
equation by a
semi-colon and
indicating the
value of the
variable means to
substitute the
number into the
equation to see if it
is a true solution.
Open Sentence Example 3
Find the solution or solutions for the equation for a given SET
of replacement values.
x  9  15;{5,6,7}
1. x  9  15
Substitute each value in the
replacement set for the variable x.
14  15
2. x  9  15
6  9  15
15  15
False when x = 5
True when x = 6
5  9  15
3. x  9  15
7  9  15
16  15
False when x = 7
The solution set for x + 9 = 15 is {6}.
Open Sentence Example 4
Find the solution or solutions for the equation for a given SET
of replacement values.
x  12  42;{56,58,60}
Substitute each value in the
replacement set for the variable x.
1. x  12  42
2. x  12  42
56  12  42
58  12  42
44  42
46  42
False when x = 56
False when x = 58
3. x  12  42
60 12  42
48  42
False when x = 60
The solution set for x - 12 = 42 is No Solution, { } for this
replacement set of {56,58,60}.
Open Sentence Example 5
Find the solution or solutions for the equation for the given
replacement set.
Substitute ALL WHOLE NUMBERS
2 x  x  x;{whole numbers}
1. 2 x  x  x
2(5)  5  5
10  10
True when x = 5
for the variable x. Try a variety of
whole numbers, some sentences have
one number that works, some have
several that work, some have none.
2. 2 x  x  x
210  10  10
20  20
True when x = 20
3. 2x  x  x
2(20)  20  20
40  40
True when x = 20
The solution of the equation 2x = x + x is true for any whole number
value. We say this sentence has infinite solutions.
Vocabulary: Inequality
• Equations are separated by equal signs.
• Mathematical sentences that have symbols separating each
side such as <, , > or  are called inequalities. The symbols
are called inequality symbols.
•
< means less than.  means less than or equal to.
•
 means greater than.  means greater than or equal to.
x  3  12
3x  24
x  5  32
2 x  1  36
Open Sentence Example 6
Find the solution or solutions for the equation for the given
replacement set.
x  3  12;{8,9,10}
Substitute each value in the
replacement set for the variable x.
1. x  3  12
8  3  12
11  12
True when x = 8
Separating an inequality by a semi-colon
and indicating the value of the variable
means to substitute the number(s) into the
inequality to see if they are a true solution.
2. x  3  12
9  3  12
12  12
False when x = 9
3. x  3  12
10  3  12
13 12
False when x = 10
The sentence is only true when x = 8. Therefore, the solution set is {8}.
Open Sentence Example 7
Find the solution or solutions for the equation for the given
replacement set.
3x  24;{7,8,9}
1. 3 x  24
Substitute each value in the
replacement set for the variable x.
2. 3 x  24
3(7)  24
21  24
3(8)  24
24  24
False when x = 7
True when x = 8
3. 3 x  24
3(9)  24
27  24
True when x = 9
The sentence is true when x = 8 and x = 9. The solution set is {8,9}.