Mathematics 10 — Exercise 2.5 · Q1 (ii)

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.

   $$2\times x^2-8\times x+6-0$$

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

   $$2\times x^2-8\times x+6$$

3. Write EVERY distributed product before collecting terms.

   $$6-8\times x+2\times x^2-0$$

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

   $$6=6$$

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

   $$-8\times x=-8\times x$$

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

   $$2\times x^2=2\times x^2$$

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.

   $$-8=-8$$

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

    $$2=2$$

11. Put the simplified groups together.

    $$2\times x^2-8\times x+6$$

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

    $$2\times x^2-8\times x+6$$

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

    $$2\times x^2-8\times x+6=0$$

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

    $$(-2-6=-8,\left(-2\right)\times \left(-6\right)=12)$$

15. Rewrite the middle term using those two parts.

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

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

    $$\left(2\times x\right)\times \left(x-1\right)-6\times \left(x-1\right)=0$$

17. The common bracket can now be factored out.

    $$2\times \left(x-3\right)\times \left(x-1\right)=0$$

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

    $$(x-3=0,x-1=0)$$

19. Take this factor equal to zero.

    $$x-3=0$$

20. Subtract the constant from both sides.

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

21. Simplify both sides.

    $$x=3$$

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

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

23. Reduce the fraction to obtain this root.

    $$x=3$$

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.

    $$2\times \left(x-3\right)\times \left(x-1\right)=0$$

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

    $$\{1,3\}$$

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

    $$6-0+2\times 0^2$$

36. Calculate this operation before combining the remaining terms.

    $$\left(-8\right)\times 0=0$$

37. Calculate this operation before combining the remaining terms.

    $$0^2=0$$

38. Calculate this operation before combining the remaining terms.

    $$2\times 0=0$$

39. Calculate this operation before combining the remaining terms.

    $$6+0+0=6$$

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

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

41. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 16=-16$$

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

    $$6-16+2\times 2^2$$

43. Calculate this operation before combining the remaining terms.

    $$\left(-8\right)\times 2=-16$$

44. Calculate this operation before combining the remaining terms.

    $$2^2=4$$

45. Calculate this operation before combining the remaining terms.

    $$2\times 4=8$$

46. Calculate this operation before combining the remaining terms.

    $$6-16+8=-2$$

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

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

48. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 32=-32$$

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

    $$6-32+2\times 4^2$$

50. Calculate this operation before combining the remaining terms.

    $$\left(-8\right)\times 4=-32$$

51. Calculate this operation before combining the remaining terms.

    $$4^2=16$$

52. Calculate this operation before combining the remaining terms.

    $$2\times 16=32$$

53. Calculate this operation before combining the remaining terms.

    $$6-32+32=6$$

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

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

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

    $$(-\infty ,1)\cup (3,\infty )$$

Answer

The final answer is shown below.