Double Inequalities
Double inequalities represent bounded regions on the number line. The double inequality a < x < b means all real numbers between a and b, where a is the smaller number and b is the larger number. All double inequalities are of the form a < x < b where one or both of the “<” signs might be replaced by “≤.” Keep in mind, though, that “ a < x < b ” is the same as “ b > x > a .” An inequality such as 10 < x < 5 is never true because no number x is both larger than 10 and smaller than 5. In other words an inequality in the form “larger number < x < smaller number” is meaningless.
The following table shows the number line region and interval notation for each type of double inequality.
Inequality 
Region on the Number Line 
Verbal Description 
Interval 
a < x < b 
All real numbers between a and b but not including a and b 
( a, b ) 

a ≤ x ≤ b 
All real numbers between a and b including a and b 
[ a, b ] 

a < x ≤ b 
All real numbers between a and b including b but not including a 
( a, b ] 

a ≤ x ≤ b 
All real numbers between a and b including a but not including b 
[ a, b ) 
Examples
Find practice problems and solutions at Double Inequalities Practice Problems  Set 1.
The Three Sides of Double Inequalities
Double inequalities are solved the same way as other inequalities except that there are three “sides” to the inequality instead of two.
Examples
Example 1:
Example 2:
Example 3:
Example 4:
Example 5:
Example 6:
Example 7:
Find practice problems and solutions at Double Inequalities Practice Problems  Set 2.
Double Inequalities with Two Variables
Double inequalities are used to solve word problems where the solution is a limited range of values. Usually there are two variables and you are given the range of one of them and asked to find the range of the other.
Examples
Example 1:
y = 3 x –2
If 7 ≤ y ≤ 10, what is the corresponding interval for x ?
Because y = 3 x – 2, replace “ y ” with “3 x − 2.”
“7 ≤ y 10” becomes “7 ≤ 3 x – 2 ≤ 10”
Example 2:
y = 4 x + 1
If – 3 < y < 3, the corresponding interval for x can be found by solving
–3 < 4 x + 1 < 3.
Example 3:
y = 3 – x
If 0 ≤ y < 4, the corresponding interval for x can be found by solving
0 ≤ 3 x < 4.
Find practice problems and solutions at Double Inequalities Practice Problems  Set 3.

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