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Double Inequalities Help

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By — McGraw-Hill Professional
Updated on Sep 26, 2011

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

Double Inequalities

All real numbers between a and b but not including a and b

( a, b )

axb

Double Inequalities

All real numbers between a and b including a and b

[ a, b ]

a < xb

Double Inequalities

All real numbers between a and b including b but not including a

( a, b ]

axb

Double Inequalities

All real numbers between a and b including a but not including b

[ a, b )

Examples

Double Inequalities 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”

Double Inequalities

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.

Double Inequalities

Example 3:

y = 3 – x

If 0 ≤ y < 4, the corresponding interval for x can be found by solving

0 ≤ 3 x < 4.

Double Inequalities

Find practice problems and solutions at Double Inequalities Practice Problems - Set 3.

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