Inequalities – Concepts and Worksheet

Concept : Remember treat inequalities just as equalities. Only when you multiply or divide, remember to flip the inequality if the number or the variable you are multiplying or dividing by is negative.

• -7x > 14

• 4x + 3 > 6x ,

• -1/2 x < ¾ ,

1. Is x > y ?

i.. wx > wy

ii. w2 x > w2 y

Concepts:

• If a < 2 and a < 7 then a < ? . Use number line to understand this concept.

• If x 2 > x then what is the range of x’s values?

• If x 2 < x , then what is range of x’s values?

• x 6 < x5 < x4 < x3 means what ?

• -1< x < 0 , Pattern of even powers X6 < x4 < x2 . pattern of odd powers x5 > x3 > x

• x even >= 0

• x odd > 0 means what ?

• If a combination of a set of variables (multiplication/division) is positive then even no of negatives are there. So if abcdef/ gh > 0 then can we say abcd/efgh > 0 ?

• If ab < 0 , or a/ b < 0 , then what can we conclude about sign of a and b ?

• If a2b3 < 0 means what ?

• If a3b5 < 0 , what can we conclude about sign of a and b ?

• If x > y , can we say x 2 > y2 ?

• If x > y then can we say x 3 > y 3 ?

• If a > b then can we say a m > b m

Easy/Medium

2. Is a2 / b > 0 ?(b not equal to zero)

i. -2 < a< 4

ii. -3 < b < 5

3. Is a2 / b < 0 ? (b not equal to zero)

i. -2 < a< 4

ii. 2 < b < 5

Medium/ Hard

4. Is a2 / b > 0 ? (b not equal to zero)

i. -2 < a< 4

ii. 2 < b < 5

5. If -1 < x < 0 , which of the following must be true ?

i. x5 < x4 ii. x 3 < 1 – x iii. x 6 < x2

A. I only B. ii only C. iii only D. I and iii only E. I, ii and iii.

6. Is a/ b > 0 ?

i. a 4 / b5 > 0

ii. ab6 > 0

7. Is bc > 0 ?

i. a2b3c5 > 0

ii. a4 / b2c5 > 0

Concepts :

• if a > b and b > c then a > c

• If a> b then

o can we say a + c > b + c ?

o can we say a –c > b – c ,

o can we say ac > bc ?

• If a > b and c > d then

o Can we say a + c > b + d ?

o Can we say a - c > b – d?

o So Remember – Inequalities can be added but not subtracted

Medium / Hard

8. Is p + q > r + s ?

i. p > r

ii. q < s

9. Is p – q > r – s ?

i. p > r

ii. q < s

10. Is a > b ?

i. a – 6 > b – 6

ii. a2 > ab

11. x = y + z , is x > y ?

i. y > 0

ii. z > 0

12. Is n < 0 ?

i. m < n

ii. – n < m

13. Is (m + z) > 0?

1. m - 3z > 0

2. 4z - m > 0

14. If x+y+z>0, is z>1?

i. z>x+y+1

ii. x+y+1 8 ?

i. d < 12

ii. d < 10

16. e < c < d

c + d + e = 24

Is c > 8 ?

i. d < 12

ii. d < 10

Concepts

Quadratic

• X2 – 6x + 9 = 0 , (x -3)^2 = 0

• X 2 – 9x + 20 = 0 , (x -5)(x -4) = 0

• X 2 – 9 x + 20 > 0 means (x -5)(x - 4) > 0 means ?

• Don’t cancel variables. Bring everything to one side and then factorise.

• Remember √4 = 2 but x2 = 4 , x = +-2

Medium / Hard

17. If x y2 = x, what is x ?

i. Y = 1

ii. Y2 not equal 1

18. What is value of x ?

i. x2 – 5x – 6 = 0

ii. x > 0

ABSOLUTE

Concepts :

• | x | = 5 means ?

• |x| > = 0 (Don’t forget 0 possibility)

• Solving |x -4| = 2

• √(x^2 = |x| (Why?)

Medium

19. If X is not equal to 0. Then √(x^2)/x = ?

A. -1, B. 0, C. 1 , D. x E. IxI / x

20. If x > 0 then √(x^2)/x =

A. -1 B. 0 C. 1 D. X E. Cannot be determined

Absolute Inequalities

• | x| > 4 means ?

• |x + 3| > 4 means ?

• |x -3| < 4 means ?

• So let’s understand this with number line.

• Remember in absolute whatever you get as answer always plug in and check if it satisfies.

• If |x| > x means what ?

• |x | = -x means what ?

• Solving mods when there are two mods on both sides. Or one mod on one side

|x -2| = |2x -2|

| x -2| = x + 3

• Remember - First use the concept that a mod is always positive – That may give u a shortcut.

Easy/Medium

21. If |a| > a then is ab > 0 ?

i) b2 > 0

ii) b3 > 0

Medium/Hard

22. Is x > 1

i) |x2 – 4| = 2x – 2

ii) X not equal to 1

23. What is the value of b?

i. |a2 – 2| = b – 2

ii. |4 – b| = 10

24. Is a > 0

i. |a + 4| = 5a - 3

ii. |a – 4| = |2a – 4|

Concepts:

• When three mods are there, use help of a number line

• E.g., |x – 2| - |x -3| = | x -4|

• |a + b| < | a| + | b |means what ?

• |a + b| = | a| + | b |means what ?

• |a – b| >| | a| – | b || means what ?

• |a – b| =| | a| – | b ||means what ?

• Equal to when they have same sign and the inequality holds true when they have opposite sign.

Hard

25. Is |a| + |b| > |a + b|

I. a2b3 < 0

II. a3b2