Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Interval: \((-1, \infty)\). Graph: Open circle at \(-1\), arrow right.

Step 1: Identify the Type of Inequality

The provided inequality is of the form \( x > -1 \), which represents a strict inequality. This means that \( x \) is greater than \(-1\) but not equal to \(-1\).

Step 2: Convert to Interval Notation

To convert the inequality \( x > -1 \) into interval notation, note that the interval starts at \(-1\) and goes to positive infinity. Since \(-1\) is not included, use a parenthesis: \(( -1, \infty )\).

Step 3: Graph the Interval on a Number Line

To graph \(( -1, \infty )\) on a number line, draw an open circle at \(-1\) to indicate it is not included in the interval. Then, draw a line or arrow extending to the right towards positive infinity.

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers between two endpoints. In the given example, the inequality \( x > -1 \) is expressed in interval notation as \(( -1, \infty )\). This representation tells us that we are considering all numbers greater than \(-1\), stretching continuously up to, but not including, infinity.

In interval notation:

• Round brackets \(( )\) are used when the endpoint is not included, called an open interval.
• Square brackets \([ ]\) are used when the endpoint is included, known as a closed interval.

For example, if the inequality were \(x \geq -1\), the interval notation would be \([-1, \infty)\) since \(-1\) would be included in the set of possible solutions.

Remember, infinities are never included, so they will always be paired with a round bracket.

Strict Inequality

A strict inequality is an expression that states one value is greater or less than another, but not equal to it. In mathematical terms, when we see a symbol like \(>\) or \(<\), it indicates a strict inequality. In our problem, the inequality \(x > -1\) signifies that \(x\) can be any number larger than \(-1\), but \(-1\) itself is not part of the solution.

Strict inequalities are crucial in determining the boundaries of a set of numbers, especially when writing in interval notation or graphing on a number line. The open or closed status of an endpoint significantly impacts the solution.

If we consider non-strict inequalities like \(\geq\) or \(\leq\), the solutions would include the boundary values, modifying how we'd express them in interval notation.

Number Line Graphing

Graphing intervals on a number line is a handy visual tool that helps clarify the range of values included in an inequality. With the inequality \(x > -1\), we translate this into the visual form \(( -1, \infty )\). Here's how to graph it:

• Identify the endpoint, which here is \(-1\). Since it’s not included in the interval, we use an open circle at \(-1\).
• Draw a line or arrow extending right from \(-1\) towards positive infinity, indicating all numbers greater than \(-1\) are part of the solution.

This graphing method provides a clear and quick reference, making it easier to interpret the inequality's solutions. It visually represents how strict inequalities exclude specific endpoints from the set of possible solutions.