Mathematics Dictionary
Dr. K. G. Shih
Inequality
Subjects
Read Symbol defintion
Q01 |
- a*x + b > 0
Q02 |
- a*x^2 + b*x + c > 0
Q03 |
- a*x^3 + b*x^2 + c*x + d > 0
Q04 |
- a*x^4 + b*x^3 + c*x^2 + d*x + e > 0
Q05 |
- a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f > 0
Q06 |
- Solve Abs(x + 2) < 4
Q07 |
- Exercises : Find graphic solutions
Q08 |
- Solve (x-1)/(x+1) > 1
Q09 |
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Q10 |
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Answers
Q01.a*x + b > 0
Graphic solutions
Equations and Functions
program 01 in Algebra 21 01
Example : F(x) = 2*x - 3, Find domain if F(x) < 0
Graphic solution
Open program 01 in Algebra 21 01
Click Menu
Click program 01 in box
Give coefficients : 2, -3
Estimate answer from diagram.
Analysis method
F(x) = 2*x - 6
If x LT -3, F(x) = (-)
If x LT +0, F(x) = (-)
If x LT +3, F(x) = (-)
If x GT +3, F(x) = (+)
Hence the domain is x less than 3 if F(x) is less than 0
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Q02. a*x^2 + b*x + c > 0
Graphic solutions
Equations and Functions
program 02 in Algebra 21 02
Example : F(x) = x^2 -6*x + 8, Find domain if F(x) < 0
Graphic solution
Open program 02 in Algebra 21 02
Click Menu
Click program 02 in box
Give coefficients : 1, -6, 8
Estimate answer from diagram.
Factor method
F(x) = x^2 - 6*x + 8 = (x - 2)*(x - 4)
If x LT +2, F(x) = (-)*(-) = (+)
If x EQ +2, F(x) = 0
If x between 2 and 4, F(x) = (+)*(-) = (-)
If x EQ +4, F(x) = 0
If x GT +4, F(x) = (+)*(+) = (+)
Hence the domain is x is between 2 and 4 if F(x) is less than 0
Go to Begin
Q03. a*x^3 + b*x^2 + c*x + d > 0
Graphic solutions
Equations and Functions
program 03 in Algebra 21 03
Example : F(x) = x^3 + 3*x^2 + 3*x + 1, Find domain if F(x) < 0
Graphic solution
Open program 03 in Algebra 21 03
Click Menu
Click program 03 in box
Give coefficients : 1, 3, 3, 1
Estimate answer from diagram.
Factor method
F(x) = x^3 + 3*x^2 + 3*x + 1 = (x + 1)^3
If x LT -1, F(x) = (-)
If x EQ -1, F(x) = 0
If x GT +0, F(x) = (+)
Hence the domain is x > 0
Go to Begin
Q04. a*x^4 + b*x^3 + c*x^2 + d*x + e > 0
Graphic solutions
Equations and Functions
program 04 in Algebra 21 04
Example : F(x) = x^4 - 7*x^2 + 1, Find domain if F(x) < 0
Graphic solution
Open program 04 in Algebra 21 04
Click Menu
Click program 04 in box
Give coefficients : 1, 0, -7, 0, 1
Estimate answer from diagram.
Go to Begin
Q05. a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f > 0
Graphic solutions
Equations and Functions
program 05 in Algebra 21 05
Example : F(x) = x^5 + x^4 - 7*x^3 - 7*x^2 + x + 1, Find domain if F(x) < 0
Graphic solution
Open program 05 in Algebra 21 05
Click Menu
Click program 05 in box
Give coefficients : 1, 1, -7, -7, 1, 1
Estimate answer from diagram.
Go to Begin
Q06. Solve |x + 2| < 4
If x > 2 and |x + 2| > 4
If x = 2 and |x + 2| = 4
If x = 0 and |x + 2| = 2
If x =-2 and |x + 2| = 0
If x =-6 and |x + 2| = |-6 + 2| = |-4| = 4
If x <-6 and |x + 2| > 4
Hence x is between -6 and 2
Go to Begin
Q07. Exercises : Find graphic solutions if F(x) > 0
1. F(x) = 2*x + 1
2. F(x) = x^2 + x + 1
3. F(x) = x^3 - 3*x^2 + 3*x - 1
4. F(x) = x^4 - 7*x^2 + 1
5. F(x) = x^5 + x^4 - 7*x^3 - 7*x^2 + x + 1
6. F(x) = x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1
Hint (x^2 + x + 1) is a factor
7. F(x) = x^7 + x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + x + 1
See Algebra Program 21 11
Hint (x + 1) and (x^2 + x + 1) are factors
Go to Begin
Q08. Solve (x-1)/(x+1) > 1
Method 1
Both sides multiply by (x+1)
Hence (x-1) > (x+1)
Both side minus x
Hence -1 > 1
There is no meaning.
Method 2 : Use asymptotes of y = (x-1)/(x+1)
If x = -1 then then y = - infinite. Hence x = -1 is asymptote.
If x = infinite, then y = (x-1)/(x+1) = 1. Hence y = 1 is asymptote.
Hence When x is greater than -1, then y is less than 1.
When x is less than -1, then y is greater than 1.
Hence (x-1)/(x+1) > 1 if x is less than -1.
Method 3 : Graphic solution
Since we know tha asymptotes : x = -1 and y = 1.
Hence we draw these two asymptotes first.
x = -1, y = -infinite.
x = 1 and y = 0.
x = 4 and y = 3/5 = 0.6
x = infinite, y = 1.
Hence we can draw curve from x = -1 to infinte and y from -infinite to 1.
When x = -1.000001, y = +infinte.
When x = -3, y = (-3 - 1)/(-3 + 1) = 2.
When x = -infinte, y = 1.
Hence we can draw curve from x = -infinite to x = -1 and y from 1 to infinite.
From graph we see that (x-1)/(x+1) > 1 if x is less than -1.
Method 4 : Sketch y = (x-1)/(x+1) by computer.
Rational function
Subject 03 program 08
y = (a*x^3 + b*x^2 + c*x + d)/(p*x^3 + q*x^2 + r*x + s)
Click start in the program
Click subject 3 in upper box
Click program 8 in lower box
Give coefficents a,b,c,d and p,q,r,s : 0, 0, 1, -1, 0, 0, 1, 1
We will see the sketch.
Go to Begin
Q09. Solve (x-1)/(x+1) > 2
Method 1
Both sides multiply by (x+1)
Hence (x-1) > 2*(x+1)
Both side minus 2*x
Hence -x - 1 > 2
Hence x > -3.
The solution is not incomplete
Method 2 : Graphic solution
Since x = -1 and y = 1 are asymptotes
Hence the solution can be obtained by draw y = (x-1)/(x+1) and y = 2.
Hence the solution : x is between -3 and -1.
Verify
When x = -3, y = (-3 - 1)/(-3 + 1) = 2
When x = -2, y = (-2 - 1)/(-2 + 1) = 3
When x = -1, y = +infinite
Go to Begin
Q10. Answer
Go to Begin
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