Mathematics 10 — Exercise 2.5 · Q1 (vi)

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.

   $$-(x^2-3\times x+2)-0$$

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

   $$-x^2+3\times x-2$$

3. Write EVERY distributed product before collecting terms.

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

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

   $$-2=-2$$

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

   $$-x^2=-x^2$$

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

   $$3\times x=3\times x$$

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.

   $$-2+0=-2$$

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.

    $$3=3$$

11. Put the simplified groups together.

    $$-x^2+3\times x-2$$

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

    $$-x^2+3\times x-2$$

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

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

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

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

15. Rewrite the middle term using those two parts.

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

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

    $$x\times \left(x-1\right)-2\times \left(x-1\right)=0$$

17. The common bracket can now be factored out.

    $$\left(x-2\right)\times \left(x-1\right)=0$$

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

    $$(x-2=0,x-1=0)$$

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.

    $$x-1=0$$

25. Subtract the constant from both sides.

    $$x-1+1=0+1$$

26. Simplify both sides.

    $$x=1$$

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

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

28. Reduce the fraction to obtain this root.

    $$x=1$$

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.

    $$-\left(x-2\right)\times \left(x-1\right)=0$$

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

    $$\{1,2\}$$

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

    $$-2-0^2+3\times 0$$

36. Calculate this operation before combining the remaining terms.

    $$0^2=0$$

37. Calculate this operation before combining the remaining terms.

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

38. Calculate this operation before combining the remaining terms.

    $$3\times 0=0$$

39. Calculate this operation before combining the remaining terms.

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

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

    $$\mathrm{test\; value}=-2$$

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

    $$\frac{3\times 1}{1\times 2}$$

42. Calculate these two products separately.

    $$\frac{3}{2}$$

43. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{2}=\frac{3}{2}$$

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

    $$-2-{\left(\frac{3}{2}\right)}^2+3\times \frac{3}{2}$$

45. Calculate this operation before combining the remaining terms.

    $${\left(\frac{3}{2}\right)}^2=\frac{9}{4}$$

46. Calculate this operation before combining the remaining terms.

    $$1\times 4=4$$

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

    $$\frac{\left(-9\right)\times 1}{1\times 4}$$

48. Calculate this operation before combining the remaining terms.

    $$\left(-9\right)\times \frac{1}{4}=-\frac{9}{4}$$

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

    $$\frac{\left(-1\right)\times 9}{1\times 4}$$

50. Calculate these two products separately.

    $$-\frac{9}{4}$$

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

    $$\frac{9\times 1}{1\times 4}$$

52. Calculate these two products separately.

    $$\frac{9}{4}$$

53. Calculate this operation before combining the remaining terms.

    $$9\times \frac{1}{4}=\frac{9}{4}$$

54. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{9}{4}=-\frac{9}{4}$$

55. Calculate this operation before combining the remaining terms.

    $$1\times 2=2$$

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

    $$\frac{9\times 1}{1\times 2}$$

57. Calculate this operation before combining the remaining terms.

    $$9\times \frac{1}{2}=\frac{9}{2}$$

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

    $$\frac{3\times 3}{1\times 2}$$

59. Calculate these two products separately.

    $$\frac{9}{2}$$

60. Calculate this operation before combining the remaining terms.

    $$3\times \frac{3}{2}=\frac{9}{2}$$

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

    $$\mathrm{LCD}=4$$

62. Calculate this operation before combining the remaining terms.

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

63. Calculate this operation before combining the remaining terms.

    $$1\times 1=1$$

64. Calculate this operation before combining the remaining terms.

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

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

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

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

    $$\frac{\left(-8\right)\times 1}{1\times 4}$$

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

    $$\mathrm{GCD}=4$$

68. Show the two divisions explicitly.

    $$\frac{-\frac{8}{4}}{\frac{4}{4}}$$

69. Evaluate the numerator division.

    $$-\frac{8}{4}=-2$$

70. Evaluate the denominator division.

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

71. Write the reduced fraction.

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

72. Calculate this operation before combining the remaining terms.

    $$\left(-8\right)\times \frac{1}{4}=-2$$

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

    $$-\frac{8}{4}$$

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

    $$-\frac{9}{4}=\frac{\left(-9\right)\times 1}{4\times 1}$$

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

    $$-\frac{9}{4}$$

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

    $$\frac{9}{2}=\frac{9\times 2}{2\times 2}$$

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

    $$\frac{18}{4}$$

78. Calculate this operation before combining the remaining terms.

    $$-8-9+18=1$$

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

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

80. Calculate these two products separately.

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

81. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{-8-9+18}{4}=\frac{1}{4}$$

83. Calculate this operation before combining the remaining terms.

    $$-2-\frac{9}{4}+\frac{9}{2}=\frac{1}{4}$$

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

    $$\mathrm{test\; value}=\frac{1}{4}$$

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

    $$-2-3^2+3\times 3$$

86. Calculate this operation before combining the remaining terms.

    $$3^2=9$$

87. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 9=-9$$

88. Calculate this operation before combining the remaining terms.

    $$3\times 3=9$$

89. Calculate this operation before combining the remaining terms.

    $$-2-9+9=-2$$

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

    $$\mathrm{test\; value}=-2$$

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

    $$[1,2]$$

Answer

The final answer is shown below.