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Title: Linear Coordinate System - Differential Calculus
Description: Hi! This is one of the topics before the actual Calculus lectures. This is part of the Pre-Calculus discussion about the Linear Coordinate System.
Description: Hi! This is one of the topics before the actual Calculus lectures. This is part of the Pre-Calculus discussion about the Linear Coordinate System.
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DIFFERENTIAL CALCULUS
Study Notes 1
“LINEAR COORDINATE SYSTEM,
ABSOLUTE VALUE, AND INEQUALITIES”
Prepared By: Tutor Win
LINEAR COORDINATE SYSTEM, ABSOLUTE VALUES, AND
INEQUALITIES
07/28/2021
In studying the fundamentals of Calculus like limits, derivatives, and
integrals, we also need to go back to the previous discussion in Algebra
wherein it demonstrates the importance of application of the basic ideas to
solve Calculus problems
...
We have also the absolute value that discusses the nature of an
integer to various computational problems
...
I know that a problem could have
more than one solution
...
Every
number we know is actually a real number
...
There are two types of a real number: a
rational, and irrational number
...
Meanwhile, irrational
is the opposite
...
See the example below
...
5
-3
0
...
Irrational Numbers
1
...
14159…
...
73205
...
” During our early maths, we have already encountered this absolute
value
...
That is,
|x| - read as ‘the absolute value of x
...
Example
|5|
=
5
|-5|
=
5
See the similarities, the absolute value of a positive and negative
integer is equal to the value of units from the origin of 0
...
So, we have 5 units! In |-5|, this means how many units we
have from the origin to the left taking 5 units also
...
LINEAR COORDINATE SYSTEM
From the term itself, we have the word “line
...
Linear coordinate system is a visual representation of real numbers
using a straight horizontal line
...
Create a point wherein you could name it as the origin – 0
...
Indicate the right side of the origin with positive values and the left
side with negative values
...
Create the points with fixed value to each other
...
5
-9
-12
-7
...
5
3
6
9
10
...
The right-side values from
it are positive while the left-side values are negative
...
The fixed value between the points is 3
...
The same context may apply to
the negative side
...
We could identify coordinate points simply as the point
...
INEQUALITIES
Inequalities are mathematical statements demonstrating a function
that have more than one solutions
...
Such inequality symbols are < (less than), > (greater than), ≥
(greater than or equal to), and ≤ (less than or equal to)
...
Solve 3x + 5 < 7
Steps:
Step 1
...
Get rid of 5 on the left side of the inequality leaving
just the term with a variable x
...
3x + 5 – 5 < 7 – 5
Step 3
...
3x < 2
Step 4
...
That is, divide both sides by 3 to get rid
of 3
...
Evaluate the inequality
...
That is, the solution set that
𝟑
satisfies the given inequality is x value greater than two𝟐
thirds
...
𝟑
4|Page
LINEAR COORDINATE SYSTEM, ABSOLUTE VALUES, AND
INEQUALITIES
07/28/2021
Example 2
...
Linear Inequality
4 < 6x – 3 < 8
Step 2
...
That is, we have to add 3 in every part of the inequality
...
Evaluate the inequality
...
In the middle, get rid of 6 to find x
...
𝟕
𝟔
<
𝟔𝒙
𝟔
<
𝟏𝟏
𝟔
Step 5
...
𝟕
𝟔
< 𝒙<
𝟏𝟏
𝟔
𝟕
𝟏𝟏
𝟔
𝟔
The final answer is < 𝒙 <
...
Or in interval notation, the answer is
)
...
Solve -3x + 9 > 1
Steps:
Step 1
...
Leave -3x on the left side because we will find x
solutions
...
To get rid of it, subtract
both sides by 9
...
Evaluate the inequality
...
To find x, divide both sides by -3
...
𝟑𝒙
𝟑
<
𝟖
𝟑
Step 5
...
𝒙 <
𝟖
𝟑
Thus, the answer is 𝒙 <
𝟖
𝟑
...
A
...
Use the interval notation in expressing your answer
...
2
...
4
...
-1 < x < 6
3 ≤ x ≤ 10
-3 ≤ x < 2
x>4
x ≤ -5
B
...
6
...
10 – 6x ≤ 2
8
...
|x + 5| = 3
10
...
1
...
Interval Notation: (-1, 6)
-1
6
2
...
Interval Notation: [3, 10]
3
10
3
...
Interval Notation: [-3, 2)
2
-3
4
...
Interval Notation: (4, ∞)
4
5
...
Interval Notation:
(-∞, −5]
-5
8|Page
LINEAR COORDINATE SYSTEM, ABSOLUTE VALUES, AND
INEQUALITIES
07/28/2021
B
...
7x – 5 ≥ 5
Step 1
...
7x – 5 + 5 ≥ 5 + 5
Step 3
...
≥
Step 5
...
10 – 6x ≤ 2
Step 1
...
10 – 6x - 10 ≤ 2 – 10
Step 3
...
≥
Step 5
...
|6x – 7| = 3
Step 1
...
|x| = -x (if x is a negative number)
Step 2
...
Solve each equation
...
6x – 7 = 3
9|Page
LINEAR COORDINATE SYSTEM, ABSOLUTE VALUES, AND
INEQUALITIES
07/28/2021
6x – 7 + 7 = 3 + 7
6x = 10
=
x=
𝟓
𝟑
b
...
|x + 5| = 3
Step 1
...
Solve each equation
...
x+5 = 3
x+5–5=3–5
x = -2
b
...
4 < 2x + 2 < 10
Step 1
...
4 - 2 < 2x + 2 – 2 < 10 – 2
Step 3
...
<
<
Step 5
Title: Linear Coordinate System - Differential Calculus
Description: Hi! This is one of the topics before the actual Calculus lectures. This is part of the Pre-Calculus discussion about the Linear Coordinate System.
Description: Hi! This is one of the topics before the actual Calculus lectures. This is part of the Pre-Calculus discussion about the Linear Coordinate System.