1.3 Open Sentences

• A mathematical statement with one or more variables is called an open sentence.

• An open sentence is neither true nor false until the variables have been replaced by specific values.

• The process of finding a value for a variable that results in a true sentence is called solving the open sentence.

Open Sentences

• This replacement value is called a solution of the open sentence.

• A sentence that contains an equals sign, = , is called an equation.

• A set of numbers from which replacements for a variable may be chosen is called a replacement set.

• A set is a collection of objects or numbers. It is often shown using braces, { }, and is usually named by a capital letter.

Open Sentences

• Each object or number in the set is called an element.

• The solution set of an open sentence is the set of elements from the replacement set that make an open sentence true.

Use a Replacement Set to Solve an Equation• Find the solution set for each equation if the

replacement set is {3, 4, 5, 6, 7}.a. 6n + 7 = 37

Since n = 5 makes the equation true, the solution 6n + 7 = 37 is 5. The solution set is {5}.

Use a Replacement Set to Solve an Equation• Find the solution set for each equation if the

replacement set is {3, 4, 5, 6, 7}.b. 5(x + 2) = 40

The solution set is {6}.

Use Order of Operations to Solve an Equation• Solve the following equation.

13 2(4)

3(5 4)q

13 8

3(1)q

21

3q 7 q

Solve Inequalities

• An open sentence that contains the symbol < , ≤ , > , or ≥ is called an inequality.

• Inequalities can be solved the same way as equations.

Find the Solution Set of an Inequality• Find the solution set for 18 – y < 10 if the

replacement set is {7, 8, 9, 10, 11, 12}.

• The solution set for 18 – y < 10 is {9, 10, 11, 12}.