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Chapter P: Prerequisite Information

Chapter P: Prerequisite Information. Section P-3: Linear Equations and Inequalities. Objectives. You will learn about: Equations Solving Equations Linear Equations in One Variable Linear Inequalities in One Variable Why: These topics provide the foundation for algebraic techniques.

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Chapter P: Prerequisite Information

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  1. Chapter P:Prerequisite Information Section P-3: Linear Equations and Inequalities

  2. Objectives • You will learn about: • Equations • Solving Equations • Linear Equations in One Variable • Linear Inequalities in One Variable • Why: • These topics provide the foundation for algebraic techniques.

  3. Vocabulary • Equation • Solution of an equation in x • Solve an equation in x • Linear equation in x • Equivalent • Linear inequality in x • Solve an inequality in x • Solution set • Solve an inequality • Double inequality

  4. Properties of Equality • Let u, v, w, and z be real numbers, variables, or algebraic expressions. • Equality Properties: • Reflexive: u = u • Symmetric: If u = v then v = u • Transitive: If u = v and v = w, then u = w • Addition: If u = v and w = z, then u + w = v + z • Multiplication: If u = v and w = z, then uw= vz

  5. Example 1:Confirming a Solution • Prove that x = -2 is a solution of the equation x3 – x + 6 = 0

  6. Linear Equation in x • A linear equation in x is one that can be written in the form ax + b = 0, where a and b are real numbers and a ≠ 0.

  7. Operations for Equivalent Equations • Two or more equations are equivalent if they have the same solution set. • An equivalent equation is obtained if one or more of the following operations is performed: • Combine like terms, reduce fractions, and remove grouping symbols. • Perform the same operation on both sides: • Addition • Subtraction • Multiplication by a non-zero constant • Division by a non-zero constant

  8. Example 2:Solving a Linear Equation • Solve: 2(2x – 3) + 3(x + 1) = 5x + 2

  9. Example 3:Solve a Linear Equation Involving Fractions • Solve:

  10. Linear Inequality in x. • A linear inequality in x is one that can be written in one of the following forms: • ax + b < 0 • ax + b ≤ 0 • ax + b > 0 • ax + b ≥ 0 • Where a and b are real numbers and a ≠ 0

  11. Properties of Inequalities • Let: • u, v, w, and z be real numbers, variables or algebraic expression • c be a real number • Transitive Property: • If u < v and v < w then u < w. • Addition: • If u < v then u + w < v + w • If u < v and w < z then u + w < v + z • Multiplication: • If u < v and c > 0, then uc < vc • If u < v and c < 0, then uc > vc

  12. Example 5:Solving a Linear Inequality • Solve:

  13. Example 5:Solving a Linear Inequality Involving Fractions • Solve and graph the solution set:

  14. Example 6:Solving a Double Inequality • Solve and graph the solution set:

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