Proofs in Solving Equations

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2.8

• Proofs in Solving Equations

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Conditional Statements

• Given a conditional statement
• Ifthen
• The if part is known as the Antecendent
• Also known as the hypothesis
• The then part is known as the Consequent
• Also known as the conclusion

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Key Questions

• If Todd plays first base, then Sam plays
  shortstop.
• What is the antecedent?
• What is the consequence?
• If Sam plays shortstop and Todd plays first base
  is this statement true?
• If Sam plays right field and Todd plays first
  base, is the statement true?

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Truth Value

• An if-then statement is only false when the
  antecedent is true and the consequence is false.

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Proving Statements
Prove that if 2x 5 gt 7, then x gt6.
Statement Reason
1. 2x 5 gt 7 1.
2. 2.
3. 3.
4. 4.
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Example

• Prove
• If 6x 7 lt 5, then x lt2

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Converse

• The converse of a conditional statement is formed
  by interchanging the antecedent and the
  consequence.
• If p, then q becomes If q, then p.
• The truth of the original if-then does not
  guarantee the truth of its converse.

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Example

• Write the converse of each statement.
• If x lt3, then x lt 4.
• If 2x 1 gt 5, then x gt 2.

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• Prove
• If x gt 2, then 5x 7 gt 3

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Example

• Solve by proving the statement and its converse.
• 7x 5 lt 19

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Example

• Solve by proving the statement and its converse.
• 6x 7 lt 37

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Equivalent Statements

• There are two kinds of steps in the derivation of
  a solution set reversible and non-reversible.
• x gt2 to x 1 gt 2 1 is reversible, the
  statements are equivalent.
• x 2 to x 2 is not reversible since x 2
  does not imply that x 2 (x could be -2)

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Theorem

• To use the addition and multiplication properties
  of equality and inequality produces equivalent
  statements under the following conditions.
• The expression added or multiplied must be
  defined for all replacements.
• The expression by which we multiply must never
  have the value 0.

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Examples

• Will each step be sure to produce an equivalent
  equation or inequality?
• Multiplying both sides by x
• Adding 2x to both sides
• Multiplying both sides by
• Adding to both sides