A kosarad üres
Number lines, intervals
Number line: a straight line on which a direction and two points are marked, to the points numbers are assigned. Thus the place of \latex{ 0 } and \latex{ 1 } is defined.
By marking the place of two distinct integers the place of \latex{ 0 } and \latex{ 1 } (thus the unit) can unambiguously be defined.
Every point of the number line corresponds to a real number and the other way round, every real number corresponds to a point on the number line. The rational and irrational numbers are both dense on the number line, i.e. there is a rational and an irrational number on any short line segment.
Number lines:
\latex{ 0 }
\latex{ 0 }
\latex{ 0 }
\latex{ 1,000 }
\latex{ 10}
\latex{ 1}
The subsets of the number line
- rays, for example:
- \latex{x} is greater than or equal to \latex{ –2 }; i.e. \latex{x} is at least \latex{ –2 }; (Figure 34/a)
- \latex{x} is greater than \latex{ 4 }; (Figure 34/b)
- \latex{x} is less than or equal to \latex{ 3.5 }; i.e. \latex{x} is at most \latex{ 3.5 }; (Figure 34/c)
- \latex{x} is less than \latex{ 2.5 }. (Figure 34/d)
Rays:
\latex{ x\geq -2 }
\latex{ x\gt 4 }
\latex{ x\leq 3.5 }
\latex{ x\lt 2.5 }
\latex{ -2 }
\latex{ 0 }
\latex{ 1 }
\latex{ 1 }
\latex{ 0 }
\latex{ 4 }
\latex{ 3.5 }
\latex{ 1 }
\latex{ 0 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2.5 }
a)
b)
c)
d)
- intervals:
- closed interval, e.g. [\latex{ –2; 3 }] the set of those real numbers for which \latex{ x } is greater than or equal to \latex{ –2 } and less than or equal to \latex{ 3 }; (Figure 34/e)
- left-closed, right-open interval, e.g. [\latex{ –2; 3 }[: \latex{ x } is greater than or equal to \latex{ –2 } and less than \latex{ 3 }; (Figure 34/f)
- left-open, right-closed interval, e.g. ]\latex{ –2; 3 }]: \latex{ x } is greater than \latex{ –2 } and less than or equal to \latex{ 3 }; (Figure 34/g)
- open interval, e.g. ]\latex{ –2; 3 }[: \latex{x} is greater than \latex{ –2 } and less than \latex{ 3 }. (Figure 34/h)
Intervals:
\latex{ -2\leq x\leq 3 }
\latex{ -2\leq x\lt 3 }
\latex{ -2\lt x\leq 3 }
\latex{ -2\lt x\lt 3 }
\latex{ -2 }
\latex{ 0 }
\latex{ 1 }
\latex{ 3 }
\latex{ 3 }
\latex{ 3 }
\latex{ 3 }
\latex{ 1 }
\latex{ 1 }
\latex{ 1 }
\latex{ 0 }
\latex{ 0 }
\latex{ 0 }
\latex{ -2 }
\latex{ -2 }
\latex{ -2 }
e)
f)
g)
h)
Figure 34
Exercises
{{exercise_number}}. Plot the following intervals on a number line.
- [\latex{ -5; 3 }]
- ]\latex{ 0; 6 }]
- [\latex{ –1; 0 }[
- ]\latex{ –4; –3 }]
- ]\latex{ 4; 5 }[
- ]\latex{ –5.5; 2 }]
- ]\latex{ 0.5; 4 }]
- [\latex{ –1; 0.5 }[
- ]\latex{ –4.5; –4 }]
- ]\latex{ 3.5; 4 }[
- [\latex{ 40; 70 }]
- [\latex{ 2,000; 5,000 }[
{{exercise_number}}. Plot the subset of the set of real numbers on a number line which corresponds to the conditions below.
- \latex{x ≤ 8}
- \latex{–1 ≤ x ≤ -0.5}
- \latex{7 \lt x}
- \latex{x \lt 1}
- \latex{x} is not less than \latex{ 3 }
- \latex{x} is not greater than \latex{ –1 }
- \latex{x} is at least \latex{ 1 } and at most \latex{ 6 }
- \latex{x} is at least \latex{ 1 } or at most \latex{ –1 }
{{exercise_number}}. Give the intervals shown on the number line with the help of interval notations.
a)
b)
c)
d)
e)
\latex{ -4 }
\latex{ 0 }
\latex{ 1 }
\latex{ 6 }
\latex{ -6 }
\latex{ 0 }
\latex{ 1 }
\latex{ 1 }
\latex{ 1 }
\latex{ 1 }
\latex{ 0 }
\latex{ 0 }
\latex{ 0 }
\latex{ 8 }
\latex{ 3 }
\latex{ 3 }
\latex{ 6 }
Let the solution sets be denoted by \latex{ A, B, C, D, E. } Give the following intervals:
\latex{A\cap B;\;A\cap D;\;B\cap C;\;A\cup C;\;B\cup C;\;A\cup D;\;A\cap C\cap D;\;D\cup E;\;B\cap E.}
{{exercise_number}}. Give the intersection of the following intervals and plot them on a number line.
- ]\latex{ –5; 1 }[ \latex{\cap} [\latex{ 1; 6 }[
- ]\latex{ –5; 1 }] \latex{\cap} [\latex{ 1; 6 }[
- ]\latex{ –5; 1 }[ \latex{\cap} ]\latex{ 1; 6 }[
- ]\latex{ –4; 3 }[\latex{\cap} [\latex{ 0; 4 }]
- [\latex{ –5; 3 }] \latex{\cap} ]\latex{ –1; 1 }[
- [\latex{ 0; 7 }[ \latex{\cap} [\latex{ –5; 3 }]
- [\latex{ –5; 3 }] \latex{\cap} [\latex{ –1; 6 }[ \latex{\cap} ]\latex{ –7; 4 }]
- [\latex{ –1; 0 }[ \latex{\cap} [\latex{ –5; 2 }] \latex{\cap} ]\latex{ –4; 7 }]
{{exercise_number}}. Give the values of \latex{ x } for which the functions represented by the following graphs have non-negative values.
\latex{ 0 }
\latex{ 1 }
\latex{ 3 }
\latex{ 5 }
\latex{ -6 }
\latex{ -4 }
\latex{ -2 }
\latex{ 0 }
\latex{ 2 }
\latex{ 4 }
\latex{ 6 }
\latex{ -6 }
\latex{ -3 }
\latex{ -1 }
\latex{ 1 }
\latex{ 3 }
\latex{ 6 }
\latex{y}
\latex{y}
\latex{y}
\latex{x}
\latex{x}
\latex{x}
{{exercise_number}}. Plot the solution of the following inequalities on a number line if the fundamental set is the set of real numbers.
- \latex{2x - 3 ≥ 5}
- \latex{4 - x ≤ 9}
- \latex{x - 1 \lt 2x }
- \latex{6x + 4 \gt 2x - 8}
- \latex{1 - 2x ≥ 3x + 16}
- \latex{7 + 5x \lt - 2x}
{{exercise_number}}. Let the solution sets of the previous exercise be denoted by \latex{ A, B, C, D, E, F } respectively. Give the following sets and plot them on the number line.
\latex{A\cap B;\;B\cap E;\;C\cap F;\;A\cap F;\;B\cup C;\;E\cap D;\;A\cap C\cap D;\;B\cap F\cap C.}
{{exercise_number}}. Which of the following statements are true and which are false?
- {\latex{x\mid x} is at least \latex{ 2 } and an integer} \latex{\cap} {\latex{x\mid x} is at most \latex{ 2 } and an integer} = {\latex{ 2 }}
- {\latex{x\mid x} is a positive real number} = {\latex{x\mid x} is at least \latex{ 0 } and a real number}
- {\latex{x\mid x} is at most \latex{ 0 } and an integer} = {\latex{x\mid x} is a negative integer}
- {\latex{x\mid x} is not less than \latex{ 0 } and is a real number} = {\latex{x\mid x} is at least \latex{ 0 } and a real number}
- {\latex{x\mid x} is a non-negative real number} = {\latex{x\mid x} is at least \latex{ 0 } and a real number}
- {\latex{x\mid x} is a non-positive real number} = {\latex{x\mid x} is at most \latex{ 0 } and a real number}
Puzzle
How is it possible to get \latex{ 1,000 } by using the number \latex{ 8 } several times and any operators?
