Загрузил Aleks Galiulin

# Higher Math G1 Ch0-927

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```Outline
1
Real Numbers
2
Inequalities and Absolute Values
3
The Rectangular Coordinate System
4
Graphs of Equations
5
Functions and Their Graphs
6
Operations on Functions
7
Trigonometri Functions
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
17 / 107
0.2 Inequalities and Absolute Values
Examples of equations
3x
17 = 6,
x2
x
6=0
x2
x
6
Examples of inequalities
3x
17 &lt; 6,
0
But of almost equal significance in calculus is the notion of solving
an inequality.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
18 / 107
0.2 Inequalities and Absolute Values
Examples of equations
3x
17 = 6,
x2
x
6=0
x2
x
6
Examples of inequalities
3x
17 &lt; 6,
0
But of almost equal significance in calculus is the notion of solving
an inequality.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
18 / 107
0.2 Inequalities and Absolute Values
Examples of equations
3x
17 = 6,
x2
x
6=0
x2
x
6
Examples of inequalities
3x
17 &lt; 6,
0
But of almost equal significance in calculus is the notion of solving
an inequality.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
18 / 107
Solutions of Inequalities
To solve an inequality, for example,
x2
x
6
0,
is to find the set of all real numbers that make the inequality true.
The solution set of an inequality is usually an entire interval of numbers or, in some cases, the union of such intervals.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
19 / 107
Solutions of Inequalities
To solve an inequality, for example,
x2
x
6
0,
is to find the set of all real numbers that make the inequality true.
The solution set of an inequality is usually an entire interval of numbers or, in some cases, the union of such intervals.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
19 / 107
Open Interval (a, b)
Several kinds of intervals will arise in our work and we introduce
special terminology and notation for them.
The inequality a &lt; x &lt; b, which is actually two inequalities, a &lt; x
and x &lt; b, describes the open interval consisting of all numbers
between a and b, not including the end points a and b.
We denote this interval by the symbol (a, b) (Figure 1). (round
bracket)
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
20 / 107
Open Interval (a, b)
Several kinds of intervals will arise in our work and we introduce
special terminology and notation for them.
The inequality a &lt; x &lt; b, which is actually two inequalities, a &lt; x
and x &lt; b, describes the open interval consisting of all numbers
between a and b, not including the end points a and b.
We denote this interval by the symbol (a, b) (Figure 1). (round
bracket)
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
20 / 107
Open Interval (a, b)
Several kinds of intervals will arise in our work and we introduce
special terminology and notation for them.
The inequality a &lt; x &lt; b, which is actually two inequalities, a &lt; x
and x &lt; b, describes the open interval consisting of all numbers
between a and b, not including the end points a and b.
We denote this interval by the symbol (a, b) (Figure 1). (round
bracket)
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
20 / 107
Closed Interval [a, b]
The inequality a  x  b describes the corresponding closed
interval, which does include the end points a and b.
This interval is denoted by [a, b] (Figure 2). (square bracket)
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Chapter 0 Preliminaries
2019-2020-01
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Possible Intervals
The table below indicates the wide variety of possibilities.
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Chapter 0 Preliminaries
2019-2020-01
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Solving Inequalities
We may perform certain operations on both sides of an inequality without changing its solution set. In particular,
1. We may add the same number to both sides of an inequality.
2. We may multiply both sides of an inequality by the same positive
number.
3. We may multiply both sides by the same negative number, but then
we must reverse the direction of the inequality sign.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
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Solving Inequalities
We may perform certain operations on both sides of an inequality without changing its solution set. In particular,
1. We may add the same number to both sides of an inequality.
2. We may multiply both sides of an inequality by the same positive
number.
3. We may multiply both sides by the same negative number, but then
we must reverse the direction of the inequality sign.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
23 / 107
Example 1
Solve the inequality 2x
set.
Higher Mathematics G1 (Dept. Math)
7 &lt; 4x
2 and show the graph of its solution
Chapter 0 Preliminaries
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Example 1
Solve the inequality 2x
set.
Higher Mathematics G1 (Dept. Math)
7 &lt; 4x
2 and show the graph of its solution
Chapter 0 Preliminaries
2019-2020-01
24 / 107
Example 2
Solve the inequality
set.
Higher Mathematics G1 (Dept. Math)
5  2x + 6 &lt; 4 and show the graph of its solution
Chapter 0 Preliminaries
2019-2020-01
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Example 2
Solve the inequality
set.
Higher Mathematics G1 (Dept. Math)
5  2x + 6 &lt; 4 and show the graph of its solution
Chapter 0 Preliminaries
2019-2020-01
25 / 107
Split Points
Before tackling a quadratic inequality, we point out that a linear
factor of the form x a is positive for x &gt; a and negative for x &lt; a.
It follows that the product (x a)(x b) can change from being
positive to negative, or vice versa, only at a or b.
These points, where a factor ((x
split points.
a) or (x
b)) is zero, are called
They are the keys to determining the solution sets of quadratic and
other more complicated inequalities.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
26 / 107
Split Points
Before tackling a quadratic inequality, we point out that a linear
factor of the form x a is positive for x &gt; a and negative for x &lt; a.
It follows that the product (x a)(x b) can change from being
positive to negative, or vice versa, only at a or b.
These points, where a factor ((x
split points.
a) or (x
b)) is zero, are called
They are the keys to determining the solution sets of quadratic and
other more complicated inequalities.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
26 / 107
Split Points
Before tackling a quadratic inequality, we point out that a linear
factor of the form x a is positive for x &gt; a and negative for x &lt; a.
It follows that the product (x a)(x b) can change from being
positive to negative, or vice versa, only at a or b.
These points, where a factor ((x
split points.
a) or (x
b)) is zero, are called
They are the keys to determining the solution sets of quadratic and
other more complicated inequalities.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
26 / 107
Split Points
Before tackling a quadratic inequality, we point out that a linear
factor of the form x a is positive for x &gt; a and negative for x &lt; a.
It follows that the product (x a)(x b) can change from being
positive to negative, or vice versa, only at a or b.
These points, where a factor ((x
split points.
a) or (x
b)) is zero, are called
They are the keys to determining the solution sets of quadratic and
other more complicated inequalities.
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
26 / 107
Example 3
Solve the quadratic inequality x 2
Higher Mathematics G1 (Dept. Math)
x &lt; 6.
Chapter 0 Preliminaries
2019-2020-01
27 / 107
Example 3
Solve the quadratic inequality x 2
Higher Mathematics G1 (Dept. Math)
x &lt; 6.
Chapter 0 Preliminaries
2019-2020-01
27 / 107
Example 3
Solve the quadratic inequality x 2
x &lt; 6.
Besides factorization, we can also use quadratic formula to find the split
points.
p
b &plusmn; b2 4ac
x=
2a
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Chapter 0 Preliminaries
2019-2020-01
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Example 3 (Con’t)
From (x
3)(x + 2) &lt; 0,
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Chapter 0 Preliminaries
2019-2020-01
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Example 4
Solve 3x 2
x
2&gt;0
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
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Example 4
Solve 3x 2
x
2&gt;0
Higher Mathematics G1 (Dept. Math)
Chapter 0 Preliminaries
2019-2020-01
29 / 107
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