Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Use inequality notation to describe the subset of real numbers.The annual rate of inflation \(r\) is expected to be at least \(3.5 \%\), but no more than \(6 \%\).

Step-by-Step Solution

Answer

The rate of inflation \(r\) can be expressed with the inequality \(0.035 \leq r \leq 0.06\).

Step 1: Define the Range

The range of possible values for the rate of inflation \(r\) is from 3.5% to 6%. We denote these two percentages as decimal fractions (0.035 and 0.06) to convert them into a form that can be used in inequality expressions.

Step 2: Use Inequality Notation

We use inequality notation to represent the range of possible values for \(r\). Since the rate must be at least 3.5% but no more than 6%, we can express this as an inequality. In terms of inequality notation, this is written as \(0.035 \leq r \leq 0.06\) where \(r\) is the annual rate of inflation.

Step 3: Interpret the Inequality

The inequality \(0.035 \leq r \leq 0.06\) means that the expected annual rate of inflation \(r\) is greater than or equal to 3.5% and less than or equal to 6%.

Key Concepts

Real Numbers

Real numbers encompass both rational and irrational numbers. They include any number you can think of on the number line. This means decimals, fractions, whole numbers, and negatives are all part of real numbers.
When we discuss inequalities with real numbers, we talk about a "range." A range is the set of all possible values that a number can take. For example, when we say the annual rate of inflation lies between 3.5% and 6%, we mean the range of real numbers expressed in this inequality.
Real numbers allow us to use decimals and fractions to describe exact measures, like inflation rates, enabling precise comparisons and calculations.

Decimal Fractions

Decimal fractions are another way of expressing fractions, except they use powers of ten. They are quite common in everyday math problems, especially in dealing with percentages.

• A decimal like 0.035 is equivalent to 3.5%. It's just another way to show the same number using place value.
• Converting a percentage to a decimal involves dividing by 100. So, 6% becomes 0.06.

This conversion is vital when you need to compare, add, or subtract percentages because decimals make arithmetic operations seamless. In the inequality notation, you use decimal fractions to set clear and standardized conditions.

Annual Rate of Inflation

The annual rate of inflation is a measure of how prices of goods and services change over a year. It's typically expressed as a percentage and indicates the rate at which purchasing power is falling.
Understanding the range of the annual rate of inflation helps in economic planning. It affects everything from household budgeting to government policy.

• If we predict inflation to be between 3.5% and 6%, it shapes expectations and decisions regarding spending and saving.
• Businesses might set product prices and wage increases in response to expected inflation rates.

By defining this rate as an inequality, such as \(0.035 \leq r \leq 0.06\), we encapsulate a realistic view of future financial conditions.