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  • 1. 2-5: Inequalities © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Use interval notation •Solve linear and compound linear inequalities •Find exact solutions of quadratic and factorable inequalities
  • 2. Important Idea In previous sections, we have been solving equalities, or equations. Now we are going to solve inequalities. The methods of solving equalities and inequalities are similar but there are important differences.
  • 3. Definition The statement c<d means that c is to the left of d on the number line. d c
  • 4. Definition The statement c>d means that c is to the right of d on the number line. d c
  • 5. Important Idea The statement c<d and d>c mean the same thing.
  • 6. Definition The statement b<c<d, called a compound inequality means: b<c and simultaneouslyc<d
  • 7. Definition ( ),c d c x d⇔ < < ( ],c d c x d⇔ < ≤ Interval Notation:Let x,c & d be real numbers with c<d: [ ],c d c x d⇔ ≤ ≤ [ ),c d c x d⇔ ≤ <
  • 8. Example Write the following using interval notation: 2 5x< < 2 5x≤ < 3 8x≤ ≤
  • 9. Try This Write the following using interval notation: 3 8x< ≤ ( ]3,8
  • 10. Try This What do you think this means? [ )19,∞ ( ),0−∞
  • 11. Important Idea Principles for solving inequalities: 1. Add or subtract the same number on both sides of the inequality.
  • 12. Important Idea Principles for solving inequalities: 2. Multiply or divide both sides of the inequality by the same positive number.
  • 13. Important Idea Principles for solving inequalities: 3. Multiply or divide both sides of the inequality by the same negative number and reverse the direction of the inequality.
  • 14. Example 2 3 5 2 11x x≤ + < + Solve. Write your answer using interval notation.
  • 15. Try This 5 2 1 7x x− ≤ − ≤ + [ ]2,8− Solve. Write your answer using interval notation.
  • 16. Example 4 3 5 18x< − < Solve. Write your answer using interval notation. Graph your answer on a number line.
  • 17. Try This Solve. Write your answer using interval notation. Graph your answer on a number line. 2 ,2 3   − ÷   2 4 3 6x− < − <
  • 18. Important Idea The solutions of the form ( ) ( )f x g x< consist of intervals on the x axis where the graph of f is below the graph of g.
  • 19. Example ( )f x ( )g x ( ) ( )f x g x<
  • 20. Important Idea The graph of ( ) ( )y f x g x= − lies above the x axis when ( ) ( )f x g x o− > and below the x axis when ( ) ( )f x g x o− <
  • 21. Example Solve: 4 3 2 10 21 40 80x x x x+ + > + Hint: Rewrite the inequality.
  • 22. Try This Solve: 4 3 2 12 4 10x x x x− − > + 2.97x < − or 4.21x >
  • 23. Important Idea Solving an inequality depends only on knowing the zeros of the function and where the graph is above or below the x- axis. The zeros are where the function touches the x axis.
  • 24. Example Find the exact solutions: 2 6 0x x− − ≤
  • 25. Example Find the exact solutions: 2 2 3 4 0x x+ − ≤ Confirm with your calculator
  • 26. Try This 2 3 2 0x x+ − ≤ Find the exact solutions: 3 17 3 17 , 2 2  − − − +    
  • 27. Example Find the exact solutions: Confirm with your calculator 6 ( 5)( 2) ( 8) 0x x x+ − − ≤
  • 28. Important Idea Steps for solving inequalities: 1. Write the inequality in one of these forms: ( ) 0f x > ( ) 0f x ≥ ( ) 0f x < ( ) 0f x ≤
  • 29. Important Idea Steps for solving inequalities: 2. Determine the zeros of f, exactly if possible, approximately otherwise.
  • 30. Important Idea Steps for solving inequalities: 3. Determine the intervals on the x axis where the graph is above or below the x axis.
  • 31. Example A store has determined the cost C of ordering and storing x laser printers. 300,000 2C x x = + The delivery truck can bring at most 450 printers. How many should be ordered to keep the cost below $1600?
  • 32. Lesson Close Tell me everything you know about solving inequalities.