What is a Double Inequality?

A double inequality is a way to express that a single expression is constrained to be within a certain range by combining two inequalities.

Understanding Double Inequalities

A double inequality is essentially a shorthand for representing two inequalities at once. Instead of writing two separate inequalities, like f(x) ≥ a and f(x) ≤ b, these can be combined into a single statement: a ≤ f(x) ≤ b. The core concept is that a single expression, f(x), is greater than or equal to a lower bound (a) and less than or equal to an upper bound (b).

Key Components:

f(x): Represents the expression being evaluated. It could be a simple variable, a more complex algebraic expression, or even a function.
a: The lower bound. This defines the minimum value f(x) can take.
b: The upper bound. This defines the maximum value f(x) can take.
≤: Symbolizes 'less than or equal to'. This is commonly used within double inequalities.
Inequalities: The combined effect creates two separate but linked statements within a single expression.

How Double Inequalities Work:

A double inequality expresses a range, or interval, within which the expression f(x) is permitted to exist. In other words, f(x) must be both greater than or equal to a and simultaneously less than or equal to b.

The inequality a ≤ f(x) asserts that f(x) is not smaller than a.
The inequality f(x) ≤ b asserts that f(x) is not larger than b.
Combined, these two statements indicate that f(x) is between a and b inclusive.

Practical Examples:

Temperature Range: A room's temperature is maintained between 20°C and 25°C. This can be represented as 20 ≤ T ≤ 25, where T is the temperature.
Weight Limits: A product's weight must fall between 500 grams and 700 grams. This is shown as 500 ≤ W ≤ 700, with W being the weight.
Grades: A student earns a score between 70 and 90 (inclusive) on a test; this can be represented by 70 <= S <= 90, where S is the score.

Solving Double Inequalities:

Solving a double inequality means finding the values of the variable (often x) that satisfy the entire combined inequality. Usually, we do this by:

Treating the inequalities separately.
Solving each inequality for the variable.
Finding the common set of values that satisfy both inequalities (the intersection of the solutions).

Benefits of Double Inequalities:

Conciseness: They provide a more compact and readable way to express bounded conditions.
Clarity: They make it clear that a value is constrained within a specific interval.
Convenience: They simplify analysis, especially in mathematics and applied fields.

Summary

As highlighted in the reference, a double inequality is a system where a single expression f(x) is bound by both a lower limit and an upper limit, usually written in the form a ≤ f(x) ≤ b. This is a concise and clear way to express that the expression f(x) is confined within a particular range.

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