D8-D10 Polynomial, rational and modulus inequalities Quiz: AQA Further Maths, Unit 2

The 20 questions

1. Solve x(x - 2)(x + 1) > 0.

   • 0 < x < 2 only
   • -1 < x < 0 or x > 2
   • x < -1 or 0 < x < 2
   • x < -1 or x > 2

2. What is the definition of |x|?

   • Always negative
   • x if x < 0, and -x if x >= 0
   • x if x >= 0, and -x if x < 0
   • x^2 for all x

3. Solve |x| < 3.

   • x = 3 or x = -3
   • x > 3 or x < -3
   • -3 < x < 3
   • x < 3 only

4. Solve x^2 - 3x - 4 < 0.

   • 1 < x < 4
   • x < -1 or x > 4
   • -1 < x < 4
   • -4 < x < 1

5. Solve (x - 1)/(x + 2) > 0.

   • x > -2 only, since the denominator is positive for every value above -2
   • x < 1 only, since the numerator is negative for every value below 1
   • -2 < x < 1, since the fraction is positive between its critical values
   • x < -2 or x > 1

6. Solve |2x - 1| <= 5.

   • -3 <= x <= 2
   • -2 <= x <= 3
   • -2 <= x < 3
   • x <= -2 or x >= 3

7. Solve |x - 2| > |x + 1|.

   • x > 2
   • x < 1/2
   • x > 1/2
   • x < -1/2

8. Solve x^3 - x > 0.

   • -1 < x < 1
   • x > 0 only
   • x < -1 or 0 < x < 1
   • -1 < x < 0 or x > 1

9. Solve x^3 - 6x^2 + 11x - 6 >= 0.

   • x <= 1 or 2 <= x <= 3
   • 1 <= x <= 3
   • 1 <= x <= 2 or x >= 3
   • x >= 2 only

10. Solve (x^2 + 1)/(x - 1) < 0.

    • x < -1
    • x < 1
    • x > 1
    • All real x

11. Solve |x + 1| + |x - 2| = 5.

    • x = 2 or x = -1
    • x = 3 only
    • x = -2 or x = 2
    • x = -2 or x = 3

12. Solve x^4 - 5x^2 + 4 < 0.

    • |x| > 1
    • -2 < x < 2
    • 1 < |x| < 2
    • |x| < 1 or |x| > 2

13. Solve 2/x < 1.

    • x = 2 only, since the fraction equals 1 at that single point
    • x < 0 or x > 2
    • x > 2 only, since the fraction is less than 1 only for large positive x
    • 0 < x < 2, since the fraction is less than 1 between 0 and 2

14. Solve |x - 1| < |2x + 1|.

    • x > 0 only, since the modulus on the right is always larger
    • x < -2 only, since the left side is larger for every value below -2
    • x < -2 or x > 0
    • -2 < x < 0, since the two modulus expressions cross at these values

15. Find the values of k for which x^2 + kx + 4 > 0 for all real x.

    • k < -4 or k > 4
    • -4 < k < 4
    • 0 < k < 4
    • k > 4 only

16. Solve x/(x + 1) >= 2.

    • -1 < x <= 2
    • x <= -2 or x > -1
    • -2 <= x < -1
    • -2 < x <= -1

17. Solve |x^2 - 4| < 5.

    • x < -3 or x > 3
    • -sqrt(5) < x < sqrt(5)
    • -3 < x < 3
    • -1 < x < 1

18. Solve (x - 2)^2 (x + 1) <= 0.

    • x <= -1 or x = 2
    • x >= -1 or x = 2
    • x <= -1 only
    • -1 <= x <= 2

19. Solve x^4 - 3x^2 - 4 > 0.

    • x < -2 or x > 2
    • x < -2 or 0 < x < 2
    • -2 < x < 2
    • x > 2 only

20. How many integer values of x satisfy |2x - 3| < 6?

    • 7
    • 9
    • 6
    • 5

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