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The Algebra, 9 classes, Section IX. THE INEQUALITY, § 50. Decision inequality with one variable
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The Characteristic inequality with variable.
1. If with one part inequality to carry in the other summand with opposite sign, that shall get the inequality, tantamount given.
2. If both parts inequality shall multiply or shall do on one and same positive number, that shall get the inequality, tantamount given.
3. If both parts inequality shall multiply or shall do on one and same negative number, having changed at sign inequality on opposite, that shall get the inequality, tantamount given.
The Numeric gap:
Inequality Geometric scene Gap
a < x < b x ∈ (a; b), a<x<b, Numeric gap, Geometric scene, Inequality ( a; b )
a ≤ x ≤ b x ∈ [a; b], a≤x≤b, Numeric gap, Geometric scene, Inequality [ a; b ]
a < x ≤ b x ∈ (a; b], a<x≤b, Numeric gap, Geometric scene, Inequality ( a; b ]
a ≤ x < b x ∈ [a; b), a≤x<b, Numeric gap, Geometric scene, Inequality [ a; b )
x ≥ a x ∈ [a; +∞), x≥a, Numeric gap, Geometric scene, Inequality [ a; +∞ )
x ≤ b x ∈ (-∞; b], x≤b, Numeric gap, Geometric scene, Inequality ( -∞; b ]
x > a x ∈ (a; +∞), x>a, Numeric gap, Geometric scene, Inequality ( a; +∞ )
x < b x ∈ (-∞; b), x<b, Numeric gap, Geometric scene, Inequality ( -∞; b )
 
Problem 68. Solve inequality:

а)   2  +  x
3
  -  3  -  x
2
  >  0 ;       
б)   3  -  y
4
  -  y  +  2
5
  ≥  2 ;
Date:   22-09-2004
Decision: а)

4  +  2x  -  9  +  3x
6
  >  0

6 > 0
5x - 5 > 0
5x > 5
x > 1

x ∈ (1; ∞), Numeric gap, Geometric scene, Inequality
Answer:   ( 1 ;   ∞ )
Date:   22-09-2004
Decision: б)

15  -  5y  -  4y  -  8
20
  ≥   2

15  -  5y  -  4y  -  8
20
  -   2   ≥   0

-9y  +  7  - 40
20
  ≥   0

20 > 0
-9y - 33 ≥ 0
-9y ≥ 33

y   ≤   33
-9

y   ≤  -3  2
3

y ∈ (-∞; -3,2/3], Numeric gap, Geometric scene, Inequality
Answer:    ( -∞ ;  -3  2
3
 ]
Date:   22-09-2004
 
Problem 69. Solve inequality:

а)   x  -  3  +  2x
2
  ≥  1  -  x
4
        б)   a  -  1
2
  -  2a  +  3
8
  >  2  +  a ;
Date:   22-09-2004
Decision: а)

x
1
  -  3  +  2x
2
  -  1  -  x
4
  ≥  0

4x  -  6  -  4x  -  1  +  x
4
  ≥   0

4 > 0
x - 7 ≥ 0
x ≥7

x ∈ [-7; ∞), Numeric gap, Geometric scene, Inequality
Answer:   [ -7 ;   ∞ )
Date:   22-09-2004
Decision: б)

a  -  1
2
  -   2a  +  3
8
  -   2  +  a
1
  >  0

4a  -  1  -  2a  +  3  -  16  -  8a
8
  >   0

8 > 0
-6a - 14 > 0
-6a > 14

a   <  -  14
6

a   <  -2  1
3

a ∈ (-∞; -2,1/3), Numeric gap, Geometric scene, Inequality
Answer:    ( -∞ ;  -2  1
3
 )
Date:   22-09-2004
 
Problem 70. Solve inequality:

а)   c  -  0,5
2
  +  c  +  0,5
2
  >  c  -  1
8
        б)   x  -  0,3
3
  -  x  -  0,6
6
  ≤  x  +  1
12
 ;
Date:   20-09-2004
Decision: а)

c  -  0,5
2
  +  c  +  0,5
2
  -  c  -  1
8
  >  0

4c  -  2  +  4c  +  2  -  c  -  1
8
  >  0

8 > 0
7c - 1 > 0
7c > 1

c  >  1
7

c ∈ (1/7; ∞), Numeric gap, Geometric scene, Inequality
Answer:  (  1
7
 ;  ∞ )
Date:   20-09-2004
Decision: б)

x  -  0,3
3
  -   x  -  0,6
6
  -   x  +  1
12
  ≤  0

4x  -  1,2  -  2x  +  1,2  -  x  +  1
12
  ≤   0

12 > 0
x + 1 ≤ 0
x ≤ -1

x ∈ (-∞; -1], Numeric gap, Geometric scene, Inequality
Answer:   ( -∞ ;   -1 ]
Date:   20-09-2004
 
Problem 73. What most natural importance n satisfies inequality:

а)  18 - 3(n - 15) > 11n  ;      б)  0,3(n - 2) < 1,2 - 0,5(n + 2)  ?
Date:   22-09-2004
Decision: а)

18 - 3n + 45 - 11n > 0
-14n + 63 > 0
-14n > -63

n   <   63
14

n < 4,5

n ∈ (-∞; 4,5), Numeric gap, Geometric scene, Inequality
n ∈ ( - ∞ ;   4,5 )

Answer:     most natural number n = 4.
Date:   22-09-2004
Decision: б)

0,3n - 0,6 -12 + 0,5n + 1 < 0
0,8n - 11,6 < 0
0,8n < 11,6

n   <   11,6
0,8

n < 14,5

n ∈ (-∞; 14,5), Numeric gap, Geometric scene, Inequality
n ∈ ( - ∞ ;   14,5 )

Answer:     most natural number n = 14.
Date:   22-09-2004
 
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