Compound Inqualities
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| Two inqualities joined by "and" or "or" are called... | Compound Inequalities
Example:
x+3<8 and x>2
2/3x>5 or -x+10<7
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| The solution set of a compound inequality formed by the word "and" is the... | Intersection of the solution of the two inequalities
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| If A={x|x is an even number greater than 0 and less than 10 and B={3,4,5,6} Find the intersection | First find A,
since x is an even number greater than 0 and less than 10 A={2,4,6,8} B={3,4,5,6}
The numbers 4 and 6 are both in the set the intersection is {4,6}
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| If A={x|x is an odd number greater than 0 and B={1,2,3,4}. Find the intersection | First find A
Since x is an odd number greater than 0 and less than 10 A={1,3,5,7,9} B={1,2,3,4}
Both numbers 1 and 3 are in the sets, The intersection is {1,3}
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| A value is a solution of a compound inequality formed by the word "and" if... | it is a solution of BOTH inequalities
Example The solution set of x is lesser than or equal to 5 and x is greater than or equal to 3 contains all values of x that make the inequalities true statements
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| Solve: 2 is less than or equal to x and x is less than or equal to 6 | Simplify 2<6x<6
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[2,6]
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| Solve: x-7<2 and 2x+1<9 | Simplify each separately
first take the first solution and add 7 to each side
x-7+7<2+7 then simplify
x<9
For the second solution
First subtract 1 from each side
2x+1-1<9-1
2x<8
Now divide both sides by 2
2x/2<8/2
x<4
Solution set is [-oo, 4]
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| The solution of a compound inequality formed by the word "or" is the | Union of the two solution sets of two inequalities
Example if A={2,4,6,8} and B={3,4,5,6}
Then {2,3,4,5,6,8} is the union
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| Solve 5x-3<10 or x+1>5 - - | Simplify each solution separately
Add 3 to both sides
5x-3+3<10+3 Then simplify
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5x<13 then divide both sides by 5
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x<13/5
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Subtract 1 from each side
x+1-1>5-1
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x>4
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the union is (-oo,13/5]U[4,oo)
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| Solve -2x-5<-3 or 6x<0 | Simplify each set separately
Add 5 to both sides
-2x-5+5<-3+5 then simplify
-2x<2 then divide both sides by -2
x>-1 or {-1,oo)
For the next solution divide both sides by 6
6x/6<0/6
x<0 or (-oo,0)
the solution set is {-oo,oo} or all real numbers
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| Tip! | When dividing a set by a negative, don't forget to reverse the inequalities!!!
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| Tip Example Solve: 3<5-x<9 | First subtract 5 from all three parts
3-5<5-x-5<9-5 Then simplify
-2<-x<4 Divide all by -1 to remove negative x(flip the inequalities)
2>x>-4 is the same as -4<x<2
[-4,2]
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