91Ó°ÊÓ

In mathematics, inequalities are used to compare different values, showing when one is less than, greater than, or otherwise. In this exercise, we're given that \(0 < a < b\). This condition lays the groundwork for proving the inequalities in the problem and indicates that both 'a' and 'b' are positive with 'b' being larger. With inequalities, one useful method is manipulating both sides by adding, subtracting, or even multiplying by a positive number without changing the nature of the inequality. However, caution is needed since multiplying or dividing by a negative quantity reverses the inequality sign.

When working with inequalities, it's helpful to remember these properties:

• Transitivity: If \(a < b\) and \(b < c\), then \(a < c\).
• Adding the same number to both sides doesn't change the inequality: If \(a < b\), then \(a+c < b+c\).
• Multiplying both sides by a positive number keeps the inequality's direction.

These rules are instrumental in proving the inequality \(a < \sqrt{ab} < b\) by manipulating expressions to derive bounding for \(\sqrt{ab}\). Understanding these fundamental properties makes solving inequalities more intuitive.

A square root of a number 'x' is a value that, when multiplied by itself, gives 'x'. In our given inequality problem, computing the square root helps in reducing the squared terms in the derived inequalities. It's crucial to note that since square roots "undo" squaring, they only work smoothly on positive numbers—just like those in our exercise condition \(0 < a < b\).

Here's what's important:

• The square root operation only affects the inequality direction if involved terms are non-negative.
• Taking square root across an inequality like \(a^2 < b^2\) is valid when values are positive, given \(0 < a < b\), leading to \(a < b\).

In the step-by-step solution, square roots help transform the inequality \(a^2 < ab < b^2\) into \(a < \sqrt{ab} < b\). This step crucially uses the property that both sides of an inequality can be square-rooted if non-negative, preserving the inequality's direction.

The reciprocal rule of inequalities is a powerful tool in comparing the inverses of positive numbers. If you're given two positive numbers such that \(0 < a < b\), their reciprocals demonstrate a reversed inequality: \(1/b < 1/a\). The intuitive reason centers on the fact that larger numbers have smaller reciprocal values.

A few core points to keep in mind:

• Reciprocal inversion flips the inequality when dealing with positive values.
• This rule relies on both numbers being positive to ensure the inequality holds in the reverse direction.
• The logic behind this is that as 'b', a larger number, increases, its reciprocal \(1/b\) decreases.

This concept underpins the demonstration in the provided solution for proving \(1/b < 1/a\). Understanding this core rule of reciprocals in the context of inequalities helps predict the behavior of fractions and inverses in broader mathematical analysis.