1. Chapter 01 – Section 05 Open Sentences

2. © William James Calhoun To solve open sentences by performing arithmetic operations. open sentence - a mathematical statement with one or more variables – the variable has to be replaced by a value solving the open sentence - the process of finding out what number the variable is holding place for solution - the replacement value for the variable in an open sentence CLARIFICATION: An open sentence in mathematics is like an English sentence with a blank in it. The sentence: __________________ is a color. is true if Blue is placed in the blank, false if Hard is placed in the blank, and neither true nor false if the blank is left empty.

3. © William James Calhoun replacement set - a set of numbers (one or more) which make an open sentence true set - a collection of objects or numbers element - each individual object in a set solution set - all the replacement values that make an open sentence become true More terms you will be using in this section:

4. © William James Calhoun The solution set is the group of numbers that “worked” when they were plugged in. Solution sets are the values that make the sentence a true statement. The numbers 4, 5, 6, 7, and 8 are all elements of this set: {4, 5, 6, 7, 8} Notice that we use brackets { } to surround sets when we list them. A replacement set is the group of numbers you will be plugging in to look for the solutions.

5. © William James Calhoun EXAMPLE 1α: Find the solution set for y + 5 < 7 if the replacement set is {0, 1, 2, 3, 4}. The solution set for y + 5  7 is {0, 1, 2}. For these problems, it will help to make a chart. Replacement set: y + 5  7? True of False? 0  7? 1  7? 2  7? 3  7? 4  7? 0 + 5 = 5 1 + 5 = 6 2 + 5 = 7 3 + 5 = 8 4 + 5 = 9 TRUE FALSE Word of Caution: The book uses “0” to answer solutions sets of {}. To say 0 is the solution set is not the same as saying {0} is the solution set. Plain 0 means there were no true replacements. {0} means the only true replacement was 0 from the replacement set.

6. © William James Calhoun EXAMPLE 1β: Find the solution set for if the replacement set is {0, 2, 4, 6, 8}. Replacement set: (x + 3)/4 >3? True of False? 0 >3? 2 >3? 4 >3? 6 >3? 8 >3?

7. © William James Calhoun EXAMPLE 2β: Solve We can use order of operations to find values for variables when we set a single variable equal to algebraic expressions. d = 5 or The solution is 5. Focus on the numerator. 5(8) Multiply. Focus on the denominator. 5(3 + 5) Parenthesis first. 40 6 + 2 Add. 3 · 2 + 2 Multiply first. 8 Numerator divided by denominator. 40 8 KEY NOTE: You can only solve for one variable with one equation.

8. © William James Calhoun EXAMPLE 2β: Solve each equation. a.b.

9. © William James Calhoun PAGE 35 #13 – 31 odd