1.6 sign charts and inequalities t

sign charts and inequalities

1. x 2 – 2x – 3 x 2 + x – 2 In factored form = (x – 3)(x + 1) (x – 1)(x + 2) So for x = –3/2 : (x – 3)(x + 1) (x – 1)(x + 2) = (–)(–) (–)(+) < 0 For x = –1/2 : (x – 3)(x + 1) (x – 1)(x + 2) = (–)(+) (–)(+) > 0 This leads to the sign charts of formulas. The sign– chart of a formula gives the signs of the outputs. Sign–Charts and Inequalities I Example B. Determine whether the outcome is x 2 – 2x – 3 x 2 + x – 2 if x = –3/2 , –1/2 . + or – for For polynomials or rational expressions, factor them to determine the signs of their outputs.

2. Example C. Let f = x 2 – 3x – 4 , use the sign – chart to indicate when is f = 0, f > 0, and f < 0 . Solve x 2 – 3x – 4 = 0 (x – 4)(x + 1) = 0  x = 4 , –1 Mark off these points on a line: ( x–4 )(x+1) + + + + + – – – – – + + + + + 4 –1 Select points to sample in each segment: Test x = – 2, get – * – = + . Hence the segment is positive. Draw + sign over it. –2 Test x = 0, get – * + = –. Hence this segment is negative. Put – over it. Test x = 5, get + * + = +. Hence this segment is positive. Put + over it. 5 Sign–Charts and Inequalities I

3. Example D. Make the sign chart of f = Select a point to sample in each segment: Test x = –3 , we've a (x – 3) (x – 1)(x + 2) The root for f = 0 is from the zero of the numerator which is x = 3. The zeroes of the denominator x = 1, –2 are the values where f is undefined (UDF). Mark these values on a real line. (x – 3) (x – 1)(x + 2) –2 1 3 UDF UDF f=0 –3 ( – ) ( – )( – ) = – segment. 2 4 Test x = 0, we've a ( – ) ( – )( + ) = + segment. Test x = 2, we've a ( – ) ( + )( + ) segment. = – Test x = 4, we've a ( + ) ( + )( + ) segment. = + – – – – + + + – – – + + + + Sign–Charts and Inequalities I

4. Example E. Solve x 2 – 3x > 4 4 –1 The solutions are the + regions: (–∞, –1) U (4, ∞) –2 5 4 –1 Note: The empty dot means those numbers are excluded. The easiest way to solve a polynomial or rational inequality is to use the sign–chart. Draw the sign–chart , sample the points x = –2 , 0, 5 (x – 4)(x + 1) + + + – – – – – – + + + + Setting one side to 0, we have x 2 – 3x – 4 > 0 or (x – 4)(x + 1) > 0. The roots are x = –1 , 4 . Sign–Charts and Inequalities I

5. Example F. Solve x – 2 2 < x – 1 3 Set the inequality to 0, x – 2 2 x – 1 3 < Put the expression into factored form, x – 2 2 x – 1 3 = (x – 2)(x – 1) 2(x – 1) – 3(x – 2) = (x – 2)(x – 1) – x + 4 Hence the inequality is (x – 2)(x – 1) – x + 4 < Draw the sign chart by sampling x = 0, 3/2, 3, 5 It has a root at x = 4, and it's undefined at x = 1, 2. 4 1 5 + + + – – + + + + – – – – 2 3/2 3 UDF UDF (x – 2)(x – 1) – x + 4 The answer are the shaded negative regions, i.e. (1, 2) U [4 ∞) . Sign–Charts and Inequalities I

6. Sign-Charts and Inequalities Exercise A. Draw the sign–charts of the following formulas. 1. (x – 2)(x + 3) 4. (2 – x)(x + 3) 5. –x(x + 3) 7. (x + 3) 2 9. x(2x – 1)(3 – x) 12. x 2 (2x – 1) 2 (3 – x) 13. x 2 (2x – 1) 2 (3 – x) 2 14. x 2 – 2x – 3 16. 1 – 15. x 4 – 2x 3 – 3x 2 (x – 2) (x + 3) 2. (2 – x) (x + 3) 3. –x (x + 3) 6. 8. –4(x + 3) 4 x (3 – x)(2x – 1) 10. 11. x 2 (2x – 1)(3 – x) 1 x + 3 17. 2 – 2 x – 2 18. 1 2x + 1 19. – 1 x + 3 – 1 2 x – 2 20. – 2 x – 4 1 x + 2

7. Sign-Charts and Inequalities Exercise B. Use the sign–charts method to solve the following inequalities. 1. (x – 2)(x + 3) > 0 3. (2 – x)(x + 3) ≥ 0 8. x 2 (2x – 1) 2 (3 – x) ≤ 0 9. x 2 – 2x < 3 14. 1 < 13. x 4 > 4x 2 (2 – x) (x + 3) 2. –x (x + 3) 4. 7. x 2 (2x – 1)(3 – x) ≥ 0 1 x 15. 2 2 x – 2 16. 1 x + 3 2 x – 2 17. > 2 x – 4 1 x + 2 5. x(x – 2)(x + 3) x (x – 2)(x + 3) 6. ≥ 0 10. x 2 + 2x > 8 11. x 3 – 2x 2 < 3x 12. 2x 3 < x 2 + 6x ≥ ≥ 0 ≤ 0 ≤ 0 ≤ 18. 1 < 1 x 2

8. C. Solve the inequalities, use the answers from Ex.1.3. Inequalities

9. (Answers to odd problems) Exercise A. 1. 3. Sign-Charts and Inequalities x = 2 x = –3 + – + x = 2 x = –3 – + – UDF 5. x = 0 x = –3 – + – 7. x = –3 + + 9. x = 1/2 x = 0 x = 3 + – + –

10. Sign-Charts and Inequalities 11. x = 1/2 x = 0 x = 3 – – + – 13. x = 1/2 x = 0 x = 3 + + + + 15. x = 0 x = -1 x = 3 + – – + 17. x = 3 x = 2 + – + UDF 19. x = -3 + – + – x = -8 x = 2 UDF UDF

11. Sign-Charts and Inequalities Exercise B. 1. ( – ∞, 3) ∪ (2, ∞) 3. [ – 3, 2] 5. ( – ∞, 3] ∪ [0, 2] 9. ( – 1, 3) 7. {0} ∪ [1/2, 3] 11. ( – ∞ , 1) ∪ (0, 3) 13. ( – ∞ , – 2) ∪ (2, ∞) 15. ( – ∞ , – 2) ∪ (2, ∞) 17. ( – 8, –2 ) ∪ (4, ∞) Exercise C. 1. The statement is not true 3. ( – ∞ , – 1 2/5) ∪ (2, ∞) 5. { 12/5} ∪ [1, ∞)