2a 2009 Modulus Inequalities

Modulus Inequalities The modulus of a real number x, written |x|, is the magnitude of x. For instance, |3| = 3 and | - 3| = 3. Using this notation, inequalities such as |x| < 2 can be written as - 2 < x < 2 and | x | > 2 can be written as x > 2 or x < - 2. Modulus inequalities can be solved by squaring both sides of the inequalities or by using graphs. It is important to remember that the earlier method (squaring both sides) is valid only when both sides of the inequality are zero or positive for all values of x, example 2 | x - 1 | < | x + 3 | or | 2x - 3 | < 12, etc. Otherwise, it is recommended to solve inequalities using the graphical method. a) | x | < 2 3 c) | 3 2 x | > 12

b) | x + 1 | <

4

3

Squaring both sides of the inequalities d) e) f) g)

| 2 4 |

2x - 3 | > 12 |x-1|<|x+3| |x| > |x-1| x + 2 | > 2x + 1

Use graphical method to solve the following inequalities |x+2| > 2x+1 i) | x + 2 | < ½ (6 - x) h)

Cross multiplying and squaring both sides of the inequalities (when denominator is positive) x |< 2;x≠-4 x+4 x +1 k) | |<4;x≠1 x −1 x2 − 4 l) | | ≤ 3; x ≠ 0 x 6 m) <|x| ; x≠±1 | x | +1 j) |

What happens when the denominator is negative? n)

| x | +1 <4;x≠±1 | x | −1

Answers 2 2 <x< ) 3 3

a)

(−

d)

(x >

15 2

;x < −

9 2

)

b)

( − 7 3 < x < 13 )

c)

(x > 8 or x < −8)

e)

( − 1 3 < x < 5)

f)

(x > 1 5 or x < − 1 3)

g) (graphical method: j)

x < −8; x > −

m)

x > 2; x < −2

8 3

k) n)

x < 1)

x<

3 5

h)

;x >

x<−

5 3

;

i)

5 3 −1<x <1

l) ;

− 10 < x <

2 3

−4 ≤ x ≤ -1 ; 1 ≤ x ≤ 4 x>

5 3