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# Compound Inqualities

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
The solution set of a compound inequality formed by the word "and" is the... Intersection of the solution of the two inequalities
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}
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}
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
Solve: 2 is less than or equal to x and x is less than or equal to 6 Simplify 2<6x<6 - - [2,6]
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]
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
Solve 5x-3<10 or x+1>5 - - Simplify each solution separately Add 3 to both sides 5x-3+3<10+3 Then simplify - 5x<13 then divide both sides by 5 - x<13/5 - Subtract 1 from each side x+1-1>5-1 - x>4 - the union is (-oo,13/5]U[4,oo)
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
Tip! When dividing a set by a negative, don't forget to reverse the inequalities!!!
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]