Mathematics 10 — Exercise 2.5 · Q1 (iii)

Full question

Solve the following inequality for real values of the variable and state its solution set.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$x^2+x-6-0$$

2. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$x^2+x-6$$

3. Write EVERY distributed product before collecting terms.

   $$-6+x+x^2-0$$

4. Calculate this individual product, including its sign and powers.

   $$-6=-6$$

5. Calculate this individual product, including its sign and powers.

   $$x=x$$

6. Calculate this individual product, including its sign and powers.

   $$x^2=x^2$$

7. Calculate this individual product, including its sign and powers.

   $$\left(-1\right)\times 0=0$$

8. These terms have the same variable part. Add ONLY their coefficients.

   $$-6+0=-6$$

9. These terms have the same variable part. Add ONLY their coefficients.

   $$1=1$$

10. These terms have the same variable part. Add ONLY their coefficients.

    $$1=1$$

11. Put the simplified groups together.

    $$x^2+x-6$$

12. Collect terms with the same power and simplify their coefficients.

    $$x^2+x-6$$

13. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$x^2+x-6=0$$

14. Split the middle coefficient into two numbers whose sum is b and whose product is a·c.

    $$(3-2=1,3\times \left(-2\right)=-6)$$

15. Rewrite the middle term using those two parts.

    $$x^2+3\times x-2\times x-6=0$$

16. Group the first two terms and the last two. Take out the common factor from each group.

    $$x\times \left(x+3\right)-2\times \left(x+3\right)=0$$

17. The common bracket can now be factored out.

    $$\left(x-2\right)\times \left(x+3\right)=0$$

18. A product is zero only if at least one factor is zero. Set each linear factor equal to zero.

    $$(x-2=0,x+3=0)$$

19. Take this factor equal to zero.

    $$x-2=0$$

20. Subtract the constant from both sides.

    $$x-2+2=0+2$$

21. Simplify both sides.

    $$x=2$$

22. Divide both sides by the coefficient of the variable.

    $$x=2\times \frac{1}{1}$$

23. Reduce the fraction to obtain this root.

    $$x=2$$

24. Take this factor equal to zero.

    $$x+3=0$$

25. Subtract the constant from both sides.

    $$x+3-3=0-3$$

26. Simplify both sides.

    $$x=-3$$

27. Divide both sides by the coefficient of the variable.

    $$x=-3\times \frac{1}{1}$$

28. Reduce the fraction to obtain this root.

    $$x=-3$$

29. Substitute this root into the original equation before simplifying.

    $$0=0$$

30. Calculate both sides; their equal values confirm the root.

    $$0=0$$

31. Substitute this root into the original equation before simplifying.

    $$0=0$$

32. Calculate both sides; their equal values confirm the root.

    $$0=0$$

33. First solve the associated equation to locate the critical values.

    $$\left(x-2\right)\times \left(x+3\right)=0$$

34. The critical values split the number line into intervals. Test one value in each interval.

    $$\{-3,2\}$$

35. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 4=-4$$

36. Substitute the test value and evaluate the expression; its sign is constant within this interval.

    $$-6-4+{\left(-4\right)}^2$$

37. Calculate this operation before combining the remaining terms.

    $${\left(-4\right)}^2=16$$

38. Calculate this operation before combining the remaining terms.

    $$-6-4+16=6$$

39. Compare the calculated value with zero using the required inequality.

    $$\mathrm{test\; value}=6$$

40. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 2}$$

41. Calculate these two products separately.

    $$\frac{1}{2}$$

42. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{2}=\frac{1}{2}$$

43. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1\right)\times 1}{1\times 2}$$

44. Calculate these two products separately.

    $$-\frac{1}{2}$$

45. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{2}=-\frac{1}{2}$$

46. Substitute the test value and evaluate the expression; its sign is constant within this interval.

    $$-6-\frac{1}{2}+{\left(-\frac{1}{2}\right)}^2$$

47. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{1}{2}\right)}^2=\frac{1}{4}$$

48. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=4$$

49. Calculate this operation before combining the remaining terms.

    $$\frac{1}{1}=1$$

50. Calculate this operation before combining the remaining terms.

    $$1\times 1=1$$

51. Calculate this operation before combining the remaining terms.

    $$\left(-6\right)\times 1=-6$$

52. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-6\times \frac{1}{1}=\frac{\left(-6\right)\times 4}{1\times 4}$$

53. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-24\right)\times 1}{1\times 4}$$

54. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=4$$

55. Show the two divisions explicitly.

    $$\frac{-\frac{24}{4}}{\frac{4}{4}}$$

56. Evaluate the numerator division.

    $$-\frac{24}{4}=-6$$

57. Evaluate the denominator division.

    $$\frac{4}{4}=1$$

58. Write the reduced fraction.

    $$-6\times \frac{1}{1}$$

59. Calculate this operation before combining the remaining terms.

    $$\left(-24\right)\times \frac{1}{4}=-6$$

60. Calculate the converted numerator; the denominator is now common.

    $$-\frac{24}{4}$$

61. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{1}{2}=\frac{\left(-1\right)\times 2}{2\times 2}$$

62. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-2\right)\times 1}{1\times 4}$$

63. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=2$$

64. Show the two divisions explicitly.

    $$\frac{-\frac{2}{2}}{\frac{4}{2}}$$

65. Evaluate the numerator division.

    $$-\frac{2}{2}=-1$$

66. Evaluate the denominator division.

    $$\frac{4}{2}=2$$

67. Calculate this operation before combining the remaining terms.

    $$\left(-2\right)\times \frac{1}{4}=-\frac{1}{2}$$

68. Calculate the converted numerator; the denominator is now common.

    $$-\frac{2}{4}$$

69. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1}{4}=\frac{1\times 1}{4\times 1}$$

70. Calculate the converted numerator; the denominator is now common.

    $$\frac{1}{4}$$

71. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 2=-2$$

72. Calculate this operation before combining the remaining terms.

    $$-24-2+1=-25$$

73. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-25\right)\times 1}{1\times 4}$$

74. Calculate these two products separately.

    $$-\frac{25}{4}$$

75. Calculate this operation before combining the remaining terms.

    $$\left(-25\right)\times \frac{1}{4}=-\frac{25}{4}$$

76. Keep the common denominator and add the signed numerators.

    $$\frac{-24-2+1}{4}=-\frac{25}{4}$$

77. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 4}$$

78. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{4}=\frac{1}{4}$$

79. Calculate this operation before combining the remaining terms.

    $$-6-\frac{1}{2}+\frac{1}{4}=-\frac{25}{4}$$

80. Compare the calculated value with zero using the required inequality.

    $$\mathrm{test\; value}=-\frac{25}{4}$$

81. Substitute the test value and evaluate the expression; its sign is constant within this interval.

    $$-6+3+3^2$$

82. Calculate this operation before combining the remaining terms.

    $$3^2=9$$

83. Calculate this operation before combining the remaining terms.

    $$-6+3+9=6$$

84. Compare the calculated value with zero using the required inequality.

    $$\mathrm{test\; value}=6$$

85. Keep the intervals with the required sign. Include roots only for ≤ or ≥; strict < or > excludes roots.

    $$(-3,2)$$

Answer

The final answer is shown below.