Mathematical Analysis I
Exercise Class 1
Roberta Raineri
3rd October 2022
1
First and Second Order Equations
Ex 1
(hint. collect like terms)
x3 + x2 − 9x − 9 = 0
Ex 2
(hint. replacement)
x4 − 5x2 + 4 = 0
Ex 3
2x2 − 4x + 5 = 0
Ex 4
Ex 5
Ex 6
3
5
− x+ >0
5
3
2
1
x−1≤− x+2
3
4
(
3x + 12 > 0
−x + 6 ≤ 3
Ex 7
10x2 + 11x + 3 > 0
1
Ex 8
−6x2 − 7x + 5 ≥ 0
2
Rational Equations and Inequalities
2.1
Exercises with Equations
Ex 1
x−1 x+2
+
=0
x−2 x+1
Ex 2
x2 + 5x + 4
=0
x3 − 4x
Ex 3
2.2
1
x+3
=1−
x−2
x
Inequalities
Recall:
A(x)
≥ (≤)0
B(x)
Strategy:
1. Put the Inequality in the general form (0 in the right part of the
inequality)
2. Study A(x) > (≥)0
3. Study B(x) > 0
4. Build the sign table and take + if > and − if <
2.3
Ex 1
Ex 2
Exercises with Inequalities
x2 + 2x − 3
≥0
1 − 7x
2x − 3
1
1
≥
+
x2 − 25
x−5 x+5
2
Ex 3
(
4x
− x1 + 4 < x+1
x(x + 1)(x2 + 1) > 0
3
x−3
x+1
Irrational Equations
Recall. The existence conditions of a general n-th root
(
p
A(x) ≥ 0 if n is even
n
A(x) →
No CE if n is odd
To solve a simple equation like
p
n
A(x) = c
after applying the CE conditions it is sufficient to compute the power of left
and right sides terms:
A(x) = cn
3.1
Ex 1
Exercises with Equations
√
x−1−
√
2x − 3 = 0
Ex 2
p
4x2 + 12x + 9 = 3
Ex 3
√
3
x+4=3
3