open sentence – a mathematical statement (sentence) that contains one or more variables, or...

•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•An open sentence is neither true nor false until the variable(s) have been replaced by intended values called replacement sets.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.

•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.

•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.

•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.

103 nNumeric Expression

Numeric Expression

Variable Expression

Variable Expression

Equations are separated by an equal sign

Equations are separated by an equal sign

3;1815 1. xxState whether the equation is true or false for the given value of the variable.

Substitute 3 into the equation for the variable x and solve.

Substitute 3 into the equation for the variable x and solve. 18315

True when x = 3True when x = 3

7;2735 2. xx

27735 Substitute 7 into the equation for the variable x and solve.


False when x = 7False when x = 7

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.


3;186 1. xxState whether the equation is true or false for the given value of the variable.


Substitute 3 into the equation for the variable x and solve. 18316


2;1532 2. xx

15232 Substitute 2 into the equation for the variable x and solve.





}7,6,5{;159 xFind the solution or solutions for the equation for the given replacement set.

Substitute each value in the replacement set for the variable x.


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.


159 1. x

1595


159 2. x1596


159 3. x1597

False when x = 7False when x = 71514 1515 1516

The solution of the equation x + 9 = 15 is x = 6

The solution of the equation x + 9 = 15 is x = 6

}60,58,56{;4212 xFind the solution or solutions for the equation for the given replacement set.





4212 1. x

421256


4212 2. x421258


4212 3. x421260

False when x = 60False when x = 604244 4246 4248

The solution of the equation x - 12 = 42 is No Solution given the replacement set {56,58,60}.

The solution of the equation x - 12 = 42 is No Solution given the replacement set {56,58,60}.

}numbers whole{;2 xxx Find the solution or solutions for the equation for the given replacement set.

Substitute any whole number value in the replacement set for the variable x.

Substitute any whole number value in the replacement set for the variable x.



xxx 2 1.

55)5(2 x


xxx 2 2. 1010102


xxx 2 3.2020)20(2

True when x = 20True when x = 201010 2020 4040

The solution of the equation 2x = x + x is true for any whole number value. This is called having many solutions or infinite solutions.

The solution of the equation 2x = x + x is true for any whole number value. This is called having many solutions or infinite solutions.

•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.

123 x 243 x 325 x 3612 x

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.


True when x = 8True when x = 8 False when x = 9False when x = 9 False when x = 10False when x = 101211 1212 1213

The solution of the equation x + 3 < 12 is true when x = 8. Therefore, the solution set for x + 3 < 12 is {8}.

The solution of the equation x + 3 < 12 is true when x = 8. Therefore, the solution set for x + 3 < 12 is {8}.

}10,9,8{;123 x

123 1. x1238

123 2. x1239

123 3. x12310

Find the solution or solutions for the equation for the given replacement set.





False when x = 7False when x = 7 True when x = 8True when x = 8 True when x = 9True when x = 92421 2424 2427

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}.

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}.

}9,8,7{;243 x

24)8(3 24)9(3 243 1. x

24)7(3 243 2. x 243 3. x

open sentence – a mathematical statement (sentence) that contains one or more variables, or...

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