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Pre - Calc

Pre - Calc. Pre-Recs P.6 Inequalities. Inequalities. Interval Notation ( greater than ) less than [ greater than or equal to ] less than or equal to. Inequalities. Interval Notation ( 3 , 9 ) Greater than 3 but less than 9. Inequality notation 3< x <9. Inequalities.

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Pre - Calc

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  1. Pre - Calc Pre-Recs P.6 Inequalities

  2. Inequalities • Interval Notation • ( greater than • ) less than • [ greater than or equal to • ] less than or equal to

  3. Inequalities Interval Notation ( 3 , 9 ) Greater than 3 but less than 9 Inequality notation 3< x <9

  4. Inequalities Interval Notation ( -1 , 4 ] Greater than -1 but less than or equal to 4 Inequality notation -1< x ≤4

  5. Write as inequality and Graph (3 , 4) [5 , 7) [-6 , 0]

  6. Bounded and Unbounded Inequalities • 2<x<3 is a bounded region on the # line (2 , 3) • X > 7 is unbounded, x can go to ∞ (7 , ∞)

  7. Write as an inequality and state if bounded (4 , 8] (-∞ , -30] [0, 4] (-∞ , ∞)

  8. Solving inequalities • Inequalities have solution sets, or solution regions • 2 inequalities that have the same solution set are said to be equivalent inequalities

  9. Solve an inequality like you solve an equation except when you multiply or divide by a negative you reverse the inequality symbol -2y + 7 > 33 -2y > 26 y < -13

  10. Double Inequality Solve them together -5 < 3x + 4 < 16 -4 -4 -4 -9 < 3x < 12 /3 /3 /3 -3 < x < 4 Check to make sure this makes sense

  11. Absolute value Inequality || < and || > or

  12. Absolute value Inequality Split into 2 inequalities a + that looks like the original ‘And’ < or ‘Or’ > Reverse the inequality and take the opposite, (negative) of the other side.

  13. Absolute value Inequality | 3x + 4| < 3 < means ‘And’ 3x + 4 < 3 And 3x + 4 > -3 -4 -4 And -4 -4 3x < -1 And 3x > -7 x < -1/3 And x > -7/3

  14. Absolute value Inequality | 2x - 6| > 4 > means ‘Or’ 2x - 6 > 4 Or 2x - 6 < -4 +6 +6 +6 +6 2x > 10 Or 2x < 2 x > 5 Or x < 1

  15. Polynomial Inequalities • A polynomial can only change signs at it’s zeros, and must remain all positive or all negative on either side of the zero • We call the zeros of a polynomial the Critical Numbersand the intervals they divide the domain into Test Intervals

  16. Polynomial Inequalities x2 – 3x - 10 < 0 Factor and find zeros (x – 5) (x + 2) = 0 Zeros are at x = 5 and x = -2 Test intervals are (-∞ , -2) ( -2 , 5) ( 5 , ∞)

  17. Lets Check Out the Graph

  18. Pick a value to test in each region (-∞ , -2) ( -2 , 5) ( 5 , ∞) -4 0 8 Evaluate the inequality at that Domain value (-4)2 - 3(-4) - 10 (0 )2 - 3(0) - 10 (8)2 - 3(8) - 10 64 - 24 - 10 - 10 16 + 12 - 10 18 - 10 30 18 not < 0 30 not < 0 - 10 is < 0

  19. The solution set will be in the region where the inequality is true x2 – 3x - 10 < 0 Is true for x values in the region ( -2 , 5)

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