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1. Open Sentences

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 • An open sentence is neither true nor false until the variable(s) have been replaced by intended values called replacement sets.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

3. 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.

4. 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.

5. 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

6. 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

7. 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

8. 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

9. 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}.

10. 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.

11. Open Sentences Vocabulary • 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. Inequality examples

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

13. 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}.