which compound inequality has no solution? a.x<5 and x<-2 b.x>5 and x<-4 c.x>5 or x<-4 d.x<5 or x<-2

Question

this is from my hw for tomorrow, i don't get it and i need help with this. thank you

Answer 1

Think about the values on a number line and if it would be possible to occur.

 

Here is our number line:

 

|––––|––––|––––|––––|––––|––––|––––|––––|––––|––––|––––|

        -4      -3     -2      -1     0       1       2       3       4      5

 

 

Let's look at choice a: x<5 and x<-2. Is it possible for x to be both less than 5 and less than -2? Yes, -3, -4, -5 etc are all values less than 5 and less than -2.

 

Choice b: x>5 and x<-4. Can x be both greater than 5 and less than -4? NO! How can x be larger than 5 AND smaller than -4? No numbers fit that description!

 

Choices c and d: x>5 or x<-4.....x<5 or x<-2 respectfully. Here it is saying "or" so it could go in either direction on the number line.

Answer 2

Note that the "and" means both conditions must be met/true while

"or" means that one or the other must work, but not necessarily both

 

a)   x<5  AND  x<-2

      Any number less than -2 will also be less than 5 so all x<-2 are solutions

 

b)  x>5 AND x<-4

     There are no numbers greater than 5 that are also less than -4

     No Solution

 

c)  x>5 OR x<-4

      All numbers that satisfy either condition work.  All x>5 and all x<-4 are valid.

       -4≤x≤5 is the only interval where this doesn't work

 

d)  x<5 or x<-2

     All x less than 5 work because they all satisfy at least one of the conditions

 

Your answer is inequality b

Answer 3

In order for a compound inequality to have a solution, the two pairs of inequalities must have a common value within their ranges.

 

 

a) 

 

x < 5      and     x < -2  

 

have many values in common.  They are negative values starting at -2. 

 

 

 

b)

 

x > 5     and     x < -4

 

have no solutions in common.  One inequality only have positive values starting with 5 as a solution. The other has negative values starting with -4 as a solution.  There is no intersection.

 

 

Your answer is (b).