Algebraic Method αိုαံုးαα္αိုαα္ ေα‘ာα္αါေαးαားαဲ့ logical rule αို αိααါαα္။
(i) ab>0 then there are two different possibilities that
(ii) Similarly, for ab<0 the possibilities may be
(i)ab>0 αိုαα္αွာ a ႏွα့္ b ေျαႇာα္ျαα္း αα္ α‘ေαါα္းαα္αိုး ျαα
္αα္။ αိုαိုαျαα
္αα္ a ႏွα့္ b αα္ α‘ေαါα္းαα္αိုး ျαα
္ααα္။ αိုαααုα္ a ႏွα့္b ႏွα
္αုαံုးαα္ α‘ႏႈα္αα္αိုး ျαα
္ααα္။
(ii) ab<0 αိုαα္αွာ a ႏွα့္ b ေျαႇာα္ျαα္း αα္ α‘ႏႈα္αα္αိုး ျαα ္αα္။ αိုαိုαျαα ္αα္ a αα္ α‘ေαါα္းαα္αိုးαု αူααွ်α္ b αα္ α‘ႏႈα္αα္αိုး ျαα ္ααα္။ αိုαααုα္ a αα္ α‘ႏႈα္αα္αိုးαု αူααွ်α္ b αα္ α‘ေαါα္းαα္αိုး ျαα ္ααα္။
Example 1
Use algebraic method, to find the solution set of the inequation 12 - 5x - 2x2 ≥ 0 and illustrate it on the number line.
Solution
12 - 5x - 2x2 ≥ 0
(4 + x)(3 - 2x) ≥ 0
α‘αα္αြα္ ေααျααဲ့ေαာ rule(i)αို αံုး၍ ေျααွα္းαါαα္။
There are two possibilities for (4 + x)(3 - 2x) ≥ 0
(i)(4 + x) ≥ 0 and (3 - 2x) ≥ 0
x ≥ -4 and x ≤ 3/2
ႏွα ္α်ိဳးαံုးαိုαူαα္ααို၊ α‘ေျαα‘ေα ႏွα ္αα္αံုးαို ေျααα္ေα ေαာ α‘ေျααα ္αုαိုαာ αူαα္။ αα္αα္αို αူαα္ αိုαα္αို number line αြဲ၍ α α₯္းα ားႏိုα္αα္။
α‘ေျαα‘ေα ႏွα
္αα္αံုးαြα္ αံုျαα
္ေαာ -4 ≤ x ≤ 3/2αိုαာ αူαါαα္။
(ii)(4 + x) ≤ 0 and (3 - 2x) ≤ 0
x ≤ -4 and x ≥ 3/2
α‘αα္αါαα္းα‘αိုα္း number line αြဲ၍ α α₯္းα ားαα္။
α‘ေျαα‘ေα ႏွα
္αα္αံုးαြα္ ေျααα္ေα
ေαာ αံုα‘ေျα ααွိαါ။
The solution set of 12 - 5x - 2x2 ≥ 0 is {x/-4 ≤ x ≤ 3/2}.
Example 2
(i)(4x + 5) ≥ 0 and (3x - 2) ≥ 0
Therefore x ≥ - 5/4 and x ≥ 2/3
The solution which satisfies both conditions is
x ≥ 2/3.
(ii)(4x + 5) ≤ 0 and (3x - 2) ≤ 0
Therefore x ≤ - 5/4 and x ≤ 2/3
The solution which satisfies both conditions is
x ≤ - 5/4.
(i) ab>0 then there are two different possibilities that
(ii) Similarly, for ab<0 the possibilities may be
(i)ab>0 αိုαα္αွာ a ႏွα့္ b ေျαႇာα္ျαα္း αα္ α‘ေαါα္းαα္αိုး ျαα
္αα္။ αိုαိုαျαα
္αα္ a ႏွα့္ b αα္ α‘ေαါα္းαα္αိုး ျαα
္ααα္။ αိုαααုα္ a ႏွα့္b ႏွα
္αုαံုးαα္ α‘ႏႈα္αα္αိုး ျαα
္ααα္။(ii) ab<0 αိုαα္αွာ a ႏွα့္ b ေျαႇာα္ျαα္း αα္ α‘ႏႈα္αα္αိုး ျαα ္αα္။ αိုαိုαျαα ္αα္ a αα္ α‘ေαါα္းαα္αိုးαု αူααွ်α္ b αα္ α‘ႏႈα္αα္αိုး ျαα ္ααα္။ αိုαααုα္ a αα္ α‘ႏႈα္αα္αိုးαု αူααွ်α္ b αα္ α‘ေαါα္းαα္αိုး ျαα ္ααα္။
Example 1
Use algebraic method, to find the solution set of the inequation 12 - 5x - 2x2 ≥ 0 and illustrate it on the number line.
Solution12 - 5x - 2x2 ≥ 0
(4 + x)(3 - 2x) ≥ 0
α‘αα္αြα္ ေααျααဲ့ေαာ rule(i)αို αံုး၍ ေျααွα္းαါαα္။
There are two possibilities for (4 + x)(3 - 2x) ≥ 0
(i)(4 + x) ≥ 0 and (3 - 2x) ≥ 0
x ≥ -4 and x ≤ 3/2
ႏွα ္α်ိဳးαံုးαိုαူαα္ααို၊ α‘ေျαα‘ေα ႏွα ္αα္αံုးαို ေျααα္ေα ေαာ α‘ေျααα ္αုαိုαာ αူαα္။ αα္αα္αို αူαα္ αိုαα္αို number line αြဲ၍ α α₯္းα ားႏိုα္αα္။
α‘ေျαα‘ေα ႏွα
္αα္αံုးαြα္ αံုျαα
္ေαာ -4 ≤ x ≤ 3/2αိုαာ αူαါαα္။(ii)(4 + x) ≤ 0 and (3 - 2x) ≤ 0
x ≤ -4 and x ≥ 3/2
α‘αα္αါαα္းα‘αိုα္း number line αြဲ၍ α α₯္းα ားαα္။
α‘ေျαα‘ေα ႏွα
္αα္αံုးαြα္ ေျααα္ေα
ေαာ αံုα‘ေျα ααွိαါ။The solution set of 12 - 5x - 2x2 ≥ 0 is {x/-4 ≤ x ≤ 3/2}.
Example 2
Find the solution set of the inequation 12x2 ≥ 10 - 7x by graphical method and illustrate it on the number line.
Solution
12x2 ≥ 10 - 7x
12x2 + 7x - 10 ≥ 0
(4x + 5)(3x - 2)≥ 0
There are two possibilities for (4 + x)(3 - 2x) ≥ 0Solution12x2 ≥ 10 - 7x
12x2 + 7x - 10 ≥ 0
(4x + 5)(3x - 2)≥ 0
(i)(4x + 5) ≥ 0 and (3x - 2) ≥ 0
Therefore x ≥ - 5/4 and x ≥ 2/3
The solution which satisfies both conditions is
x ≥ 2/3.
(ii)(4x + 5) ≤ 0 and (3x - 2) ≤ 0
Therefore x ≤ - 5/4 and x ≤ 2/3
The solution which satisfies both conditions isx ≤ - 5/4.
The solution set of 3x2 < x2 - x + 3 is {x/ x ≤- 5/4 or x ≥ 2/3}.
α
ာαα်αူ၏ α‘αြα်αို αေးα
ားα
ွာα
ောα့်αျှော်αျα်!
Chapter 4
ReplyDeleteAlgebraic method αိုαွα္းျααာ ေαာ့αုα္αါαα္
Example (2 ) ေαးαြα္းαွာ by graphical method αိုαေαးαားαΏαီး αြα္ေαာ့ algebraic method ျαα ္ေααα္ ။ေαးαြα္းαိုα္αာ αွားαြားαα္ αα္αါαα္။
Post a Comment