Open Sentences

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Open Sentences
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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  the variable(s) have been replaced by intended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.

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Open Sentences
Vocabulary

• Solving the open sentence Finding a value from
  the replacement set that will make the open
  sentence a true statement. An open sentence may
  have more than one solution.
• Solution The replacement number that actually
  makes the equation a true statement. An open
  sentence may have one solution, several
  solutions, or no solutions.

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Open Sentences
Vocabulary

• Set A collection of objects or numbers. Sets
  are represented by using braces .
• Element 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.

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Open Sentences
Vocabulary

• 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
Numeric Expression
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Open Sentences
Open sentence example
State whether the equation is true or false for
the given value of the variable.
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.
Substitute 3 into the equation for the variable
x and solve.
True when x 3
Substitute 7 into the equation for the variable x
and solve.
False when x 7
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Open Sentences
Open sentence example
State whether the equation is true or false for
the given value of the variable.
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.
Substitute 3 into the equation for the variable
x and solve.
True when x 3
Substitute 2 into the equation for the variable x
and solve.
False when x 2
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Open Sentences
Open sentence example
Find the solution or solutions for the equation
for the given replacement set.
Separating an equation by a semi-colon and
indicating the value of the variable means to
substitute the number(s) into the equation to see
if they are a true solution.
Substitute each value in the replacement set for
the variable x.
False when x 7
False when x 5
True when x 6
The solution of the equation x 9 15 is x 6
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Open Sentences
Open sentence example
Find the solution or solutions for the equation
for the given replacement set.
Separating an equation by a semi-colon and
indicating the value of the variable means to
substitute the number(s) into the equation to see
if they are a true solution.
Substitute each value in the replacement set for
the variable x.
False when x 60
False when x 58
False when x 56
The solution of the equation x - 12 42 is No
Solution given the replacement set 56,58,60.
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Open Sentences
Open sentence example
Find the solution or solutions for the equation
for the given replacement set.
Separating an equation by a semi-colon and
indicating the value of the variable means to
substitute the number(s) into the equation to see
if they are a true solution.
Substitute any whole number value in the
replacement set for the variable x.
True when x 20
True when x 5
True when x 20
The solution of the equation 2x x x is true
for any whole number value. This is called having
many solutions or infinite solutions.
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Open Sentences
Vocabulary

• Equations are separated by equal signs.
• Mathematical sentences that have symbols
  separating each side such as lt, ?, ?, or ? are
  called inequalities. The symbols are called
  inequality symbols.
• lt means less than, ? means less than or equal
  to, ? means greater than, ? means
  greater than or equal to.

Inequality examples
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Open Sentences
Open sentence example
Find the solution or solutions for the equation
for the given replacement set.
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.
Substitute each value in the replacement set for
the variable x.
False when x 10
True when x 8
False when x 9
The solution of the equation x 3 lt 12 is true
when x 8. Therefore, the solution set for x 3
lt 12 is 8.
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Open Sentences
Open sentence example
Find the solution or solutions for the equation
for the given replacement set.
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.
Substitute each value in the replacement set for
the variable x.
True when x 9
True when x 8
False when x 7
The solution of the equation 3x ? 24 is true when
x 8 and x 9. Therefore, the solution set for
3x ? 24 is 8,9.