How do you find the numbers such that four less than half the number is at least five and at most ten?

2 Answers
Oct 29, 2016

The number is any value between #18# and #28#, inclusive.

Explanation:

"#4# less than half the number" is written algebraically as :

#x/2 - 4#.

Since this expression "is at least #5#", it is greater than or equal to #5#, and since it is "at most #10#", it is less than or equal to #10#. Putting all this together gives the compound inequality:

#5 <= x/2 -4 <= 10#

Remember that to simplify this type of inequality, your objective is to isolate the variable in the center section of the statement. We will accomplish this by adding #4# to each section and then multiplying the entire inequality by #2#.

#5 + 4 <= x/2 - 4 + 4 <= 10 + 4#

#9 <= x/2 <= 14#

#2(9 <= x/2 <= 14)#

#18 <= x <= 28#

Therefore the number can be any number between #18# and #28#, inclusive.

Oct 29, 2016

#18<=x<=28# i.e #x# is between #18# and #28#

Explanation:

Let the number be #x#,

then four less than half the number is #x/2-4#

and as it is at least #5#, we have #x/2-4>=5#

or #x/2>=5+4# i.e. #x/2>=9# and #x>=18#

Further as #x/2-4# is at most #10#, we have #x/2-4<=10#

or #x/2<=10+4# i.e. #x/2<=14# and #x<=28#

Combining the two we have #18<=x<=28#

i.e #x# is between #18# and #28#