Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-20: Express the interval as an inequality in the variable \(x\). $$ [-4,-1] $$

Step-by-Step Solution

Answer

\(-4 \leq x \leq -1\)

Step 1: Understand the Interval Notation

The interval \([-4, -1]\) is a closed interval, which means it includes the endpoints \(-4\) and \(-1\). This is indicated by the square brackets.

Step 2: Express as a Double Inequality

For a closed interval \([-4, -1]\), the variable \(x\) satisfies the condition \(-4 \leq x \leq -1\). This means \(x\) can be any value between \(-4\) and \(-1\), including \(-4\) and \(-1\).

Key Concepts

Interval Notation

Interval notation is a way to describe a range of numbers. It's like a shorthand that shows where a set of numbers starts and ends. This notation can be used to represent both closed and open intervals. When you see something like \([-4, -1]\), you're looking at an interval that includes all numbers between -4 and -1, and it also includes the endpoints, -4 and -1. The square brackets \([]\) are key here, showing that the boundary numbers are part of the set.

• If the brackets are square (\([]\)), it indicates a closed interval.
• If the brackets are round (\(()\)), it indicates an open interval, meaning the endpoints are not included.

Understanding interval notation is essential for reading and writing mathematical conditions easily. It simplifies the communication of which numbers are included in an interval without having to spell out each possible value.

Closed Interval

A closed interval is a specific type of interval in mathematics. It includes the boundary numbers along with all the numbers in between them. For mathematical notation, closed intervals are represented with square brackets.Placing a square bracket means, "Include this number!"In a closed interval like \([-4, -1]\), both -4 and -1 are included. This means that if you were marking numbers on a number line, you would highlight everything from -4 through -1, not skipping any number.
Closed intervals are often used in mathematical expressions and functions where it is necessary to include the edge points in calculations. They are fundamental in calculus and other areas of mathematics that deal with continuous ranges of values.

Double Inequality

A double inequality is a compact way to express two inequalities at once, showing a range of values that a variable can take. In our example, the closed interval \([-4, -1]\) converts to the double inequality \(-4 \leq x \leq -1\).This tells us that \(x\) is a number that is greater than or equal to -4 but less than or equal to -1. The double inequality is read from left to right:

• \(-4 \leq x\) means \(x\) is at least -4.
• And \(x \leq -1\) means \(x\) is at most -1.

Double inequalities are extremely helpful for quickly conveying the range of acceptable values for a variable, especially when working through algebraic equations or geometric problems.