Problem 47 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$(-3, \infty)$$

Step-by-Step Solution

Answer

The inequality is \(x > -3\). The interval starts at \(-3\) and extends to infinity.

Step 1: Understanding Interval Notation

The interval \((-3, \infty)\) is written in interval notation where \(-3\) is not included (open interval) and goes to positive infinity implying that it includes all numbers greater than \(-3\). The parentheses indicate that \(-3\) is not part of the interval.

Step 2: Express in Inequality Notation

Inequality notation describes the range of values included within the interval. For the interval \((-3, \infty)\), this is expressed as:\[ x > -3 \]This means \(x\) represents any number greater than \(-3\).

Step 3: Plotting on a Number Line

To show \((-3, \infty)\) on a number line, draw an open circle at \(-3\) to indicate \(-3\) is not included, then draw a line extending to the right, which represents all numbers greater than \(-3\). There is no endpoint because the interval extends to infinity.

Key Concepts

Inequality Notation

Inequality notation is a way to express the range of values an unknown variable can take. It uses mathematical symbols like <, >, ≤, and ≥. With inequalities, we showcase which side of a given value a variable lies.

• '< ' or '> ' are used when a value is not included in the solution set.
• '≤ ' or '≥ ' are applied when a value is included in the solution set.

Where interval notation uses parentheses and brackets, inequality notation uses these symbols to provide a clear mathematical expression of a condition. For instance, the inequality notation for the interval (-3, ∞) is expressed as \(x > -3\). This means all values of \(x\) greater than -3 are considered valid.

Number Line

A number line is a visual representation that helps understand numeric intervals and inequalities. It is a crucial tool in mathematics for illustrating the range of numbers.

• The number line consists of a horizontal line with numbers placed at equal intervals along it.
• You can use it to show integers, fractions, and decimals, creating a broader picture of numerical relationships.
• On a number line, circles or dots can indicate specific numbers.

For the interval \((-3, \infty)\), the number line uses an open circle at -3 to show that this point is not included. A line extends to the right from this circle, illustrating the inclusion of all numbers greater than -3, stretching towards infinity.

Open Interval

An open interval in mathematics does not include its endpoints. It is denoted using parentheses, like (-3, 5). This notation tells us someone is talking about numbers strictly between -3 and 5—neither -3 nor 5 is part of the interval.Understanding open intervals is essential:- They differ from closed intervals, which include their endpoints and use square brackets like \([3, 5]\).- An open interval, such as (-3, ∞), shows that the numbers from -3 and beyond, reaching towards infinity, are included, but -3 itself is not.- In our example (-3, ∞), only the values greater than -3 are part of the solution, demonstrated by the open circle on the number line.