Sheet 1. Equations, Inequalities and Sets

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1 Equations, Inequalities and Sets

Sheet 1. Equations, Inequalities and Sets

Exercise 1.1. Solve the given equations.

a) (x − 3)(x + 5) = 0 b) (2 − x)(x + 3) = 0 c) x²+ 8x + 12 = 0 d) x²+ x + 1 = 0 e) x²− x − 30 = 0 f) x(x − 2) = 3(x − 2) g) (3x + 2)² = 7(3x + 2) h) x⁴− 10x²+ 9 = 0 i) |2x − 5| = 3

j) 2^2x= 32 k) 4^x− 3 · 2^x− 4 = 0 l) log2x = 0 m) logx8 = 1

8 m) log2(x − 4) = 3 o) (log x)³− log x = 0 Exercise 1.2. Solve the given inequalities.

a) 4 ≤ 3x − 2 < 13 b) 2x + 1 ≤ 4x − 3 ≤ x + 7 c) |x − 5| < 2 d) |3x + 2| ≥ 4 e) x²− 5x + 6 ≥ 0 f) x²+ 2x + 1 ≤ 0 g) x³+ 3x²− 4x > 0 h) 2^x < 32 i) 1

4

4x

< 1 64

j) 7^x≤ 1 k) 1

3

_3x−2^x−3

≤ 3 l) log¹

2(x + 1) > 3 m) log3(x²+ 2) > 3 n) log2(x + 1) + log₂(x − 1) < 3

Exercise 1.3. For given sets A and B nd A ∪ B, A ∩ B, A \ B. Mark the results on the real axis.

a) A = {x ∈ R : x²+ 8x + 12 < 0} B = {x ∈ R : (x − 2)(x + 1) ≥ 0}

b) A =

x ∈ R : x < 1 x

B =

x ∈ R : 1 + x 1 − x > 1

c) A = {x ∈ R : 2x²+ x ≤ 1} B = {x ∈ R : |x − 1| > 2}

d) A = {x ∈ R : ||x + 1| + 2| = 2} B = {x ∈ R : x²+ x > 1}

e) A = {x ∈ R : x² < 3} B = {x ∈ R : |x + 2| > 3}

f) A =

x ∈ R : x²− 2x x²− 4 = 0

B = {x ∈ R : |x − 1| ≤ 5}

g) A = {x ∈ R : x³+ 3x < 4x²} B = {x ∈ R : (x + 1)(x − 2)(x + 3) ≥ 0}

h) A = {x ∈ R : (2 − x)(x + 1)(x + 3) ≥ 0} B = {x ∈ R : x³− x² ≤ 0}

i) A =

x ∈ R : 1 x < 4

B =

x ∈ R : 4 x < x

j) A =

x ∈ R : x²− 1 x²+ 1 ≥ 0

B =x ∈ R : x + ¹₂

< 1 k) A = {x ∈ R : 2^x+1 > 4} B =n

x ∈ R : ¹₃2x

< 9o

Last update: November 12, 2008 1

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1 Equations, Inequalities and Sets

l) A =n

x ∈ R : 7^3x−5^x+2 ≥√ 7o

B =n

x ∈ R : ¹₃^x+3_1−x

> 1o

m) A = {x ∈ R : 2^x+1+ 2^x−1 ≤ 20} B = {x ∈ R : 5^x+ 3 · 5^x−2 > 28}

n) A = {x ∈ R : 4^x+1+ 2^2−2x < 17} B =

x ∈ R : x x + 1 > 3

o) A = {x ∈ R : log2(x²− 1) < 3} B = {y ∈ R : 0 < |y| < 3}

p) A =n

x ∈ R : log¹₂(x + 5) > 0 o

B = {x ∈ R : log(x + 2) > 0}

q) A = x ∈ R : log√

x + 1 = 2

B =n

x ∈ R : ¹₂^1−x_x

≤ 1o r) A = {x ∈ R : logx(x + 1) > 1} B =

x ∈ R : 2 + x 3 − x ≤ 1

Exercise 1.4. In the Cartesian (rectangular) coordinate system mark the sets A × B and B × A, where:

a) A = [1, 5] B = [2, 3] b) A = [1, 5] B = [2, ∞] c) A = [1, 4] ∪ (5, 7) B = (1, 2]

d) A = {2} B = (1, 3) e) A = (∞, −1) B = [1, ∞) f) A = R B = {3}

g) A = R B = {3} ∪ (4, 5) h) A = {1, 2, 3} B = {4, 5}

Last update: November 12, 2008 2