Problem 55 - Precalculus Mathematics for Calculus

Question

Express the inequality in interval notation, and then graph the corresponding interval. $$x>-1$$

Step-by-Step Solution

Answer

Interval notation: \((-1, \infty)\); graph uses an open circle at \(-1\) and a ray to the right.

Step 1: Understand the Inequality

The given inequality is \(x > -1\). This means that \(x\) can take any value greater than \(-1\), but not \(-1\) itself.

Step 2: Write the Inequality in Interval Notation

To express \(x > -1\) in interval notation, we use an open interval because \(-1\) itself is not included. Therefore, the interval notation is \((-1, \infty)\).

Step 3: Graph the Interval on a Number Line

Draw a number line, and locate \(-1\) on it. Use an open circle to indicate that \(-1\) is not included in the interval. Draw a ray extending to the right from \(-1\) to show that all numbers greater than \(-1\) are included.

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers along a number line. It uses parentheses and brackets to show intervals of inclusivity and exclusivity. For the inequality \( x > -1 \), we write it as \((-1, \infty)\).
This means:

• The parenthesis \((-1\) indicates that \(-1\) is not included in the set of solutions.
• \(\infty)\) symbolizes that there is no upper limit to the values \(x\) can take, extending to positive infinity.
• Parentheses are always used with infinity because infinity is not a number, so it cannot be "included".

Interval notation is a succinct way of expressing ranges and is often used in higher mathematics because of its precision and clarity.

Number Line

A number line helps visualize intervals and inequalities. When representing inequalities like \(x > -1\), we place the numbers on a horizontal line.
Here's how you do it:

• First, identify the critical number in the inequality, which is \(-1\) in this case.
• Mark \(-1\) on the number line.
• Use an open circle on \(-1\) because the inequality \(x > -1\) means \(-1\) itself is not included.
• Shade or draw a ray to the right, starting from the open circle, indicating that all numbers greater than \(-1\) are part of the solution.

The number line is a powerful tool for visually understanding the concept of inequalities and how they relate to intervals.

Inequality Graphing

Graphing an inequality provides a visual representation that is easy to understand. To graph \(x > -1\) on a number line, first identify that \(-1\) is where the inequality changes, but it is not part of the solution set.

• Place an open circle on \(-1\) to graphically show that it is excluded from the solution.
• Draw an arrow extending to the right, starting from the open circle.
• The arrow indicates that every number greater than \(-1\) is included in the solution set, highlighting infinity.

This method provides a quick, clear way to communicate which numbers satisfy \(x > -1\) and to convey the concepts behind the inequality visually.