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P3: Solving linear equations and linear inequalities

P3: Solving linear equations and linear inequalities. Algebraic Properties. Properties of Equality (let u, v, w, and z be real numbers, variables, or algebraic expressions). 1. Reflexive. u = u. 2. Symmetric. If u = v, then v = u. 3. Transitive. If u = v, and v = w, then u = w.

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P3: Solving linear equations and linear inequalities

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  1. P3: Solving linear equations and linear inequalities

  2. Algebraic Properties Properties of Equality (let u, v, w, and z be real numbers, variables, or algebraic expressions) 1. Reflexive u = u 2. Symmetric If u = v, then v = u 3. Transitive If u = v, and v = w, then u = w 4. Addition If u = v and w = z, then u + w = v + z 5. Multiplication If u = v and w = z, then uw = vz

  3. Solving Equations A linear equation in x is one that can be written in the form ax + b = 0 where a and b are real numbers with a = 0 A solution of an equation in x is a value of x for which the equation is true. • So, how many solutions are there to a linear equation in one variable???

  4. Let’s practice… Solve for the unknown:

  5. Let’s practice… Solve for the unknown: 8 8

  6. Let’s practice… Solve for the unknown and support with grapher: Now, how do we get graphical support???

  7. Linear Inequality in x A linear inequality in x is one that can be written in the form ax + b < 0, ax + b < 0, ax + b > 0, or ax + b > 0 where a and b are real numbers with a = 0 A solution of an inequality in x is a value of x for which the inequality is true. The set of all solutions of an inequality is the solution set of the inequality.

  8. Properties of Inequalities Let u, v, w, and z be real numbers, variables, or algebraic expressions, and c a real number. 1. Transitive If u < v and v < w, then u < w 2. Addition If u < v, then u + w < v + w If u < v and w < z, then u + w < v + z 3. Multiplication If u < v and c > 0, then uc < vc If u < v and c < 0, then uc > vc (the above properties are true for < as well – there are similar properties for > and >)

  9. Guided Practice: Solve the inequality: When solving inequalities, don’t forget to switch the inequality sign whenever you multiply or divide by a negative number!!!

  10. Guided Practice: Solve the inequality, write your answer in interval notation, and graph the solution set: –2 0

  11. Guided Practice Solve the double inequality, write your answer in interval notation, and graph the solution set: (–7, 5] –7 0 5

  12. Whiteboard Practice: Solve the inequality, write your answer in interval notation, and graph the solution set: –11/7 0 11 8 – , – 7

  13. Whiteboard Practice: Solve the inequality:

  14. Whiteboard Practice: Solve and support with grapher: 12 12 Graphical Support?

  15. Homework: • p. 28-29 11-27 odd, 35-53 odd

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