Problem 51 - Algebra and Trigonometry

Question

\(47-52\) : 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 \(-1\) with arrow to the right.

Step 1: Understand the Inequality

The given inequality is \( x > -1 \). This means that \( x \) can be any number greater than \(-1\). The inequality symbol '>' indicates that \(-1\) is not included in the solution set.

Step 2: Write in Interval Notation

Since \( x \) needs to be greater than \(-1\), but not equal to it, we start the interval right after \(-1\). The interval continues to infinity because there is no upper limit on \( x \). In interval notation, this is expressed as \((-1, \infty)\). Here, a parenthesis '(', indicates that \(-1\) is not included, and an infinity symbol indicates an unbounded interval in the positive direction.

Step 3: Graph the Interval

To graph \((-1, \infty)\), first draw a number line. Place a point at \(-1\) and draw an open circle around it; an open circle indicates that \(-1\) is not included in the interval. Draw a line or arrow extending to the right from \(-1\) to indicate that all numbers greater than \(-1\) are included. Keep extending the line or arrow indefinitely to the right, representing infinity.

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand used to describe a range of numbers. It uses brackets or parentheses to indicate whether endpoints are included or excluded in the interval.

• A parenthesis '(', ')' indicates that the endpoint is not included (also known as an open interval).
• A bracket '[', ']' indicates that the endpoint is included (a closed interval).

Consider the inequality from the exercise: \( x > -1 \). In this case:

• Since \( x \) is greater than \(-1\) but does not include \(-1\), we use a parenthesis: \((-1\).
• There is no upper limit given, so \( x \) goes to infinity. We represent this with \( \infty \) and a parenthesis because infinity is not a number that can be included: \( ( \infty )\).

Putting it all together, the interval notation for \( x > -1 \) is \((-1, \infty)\). This communication is concise and clear, showing the full range of possible values for \( x \).

Graphing Inequalities

Graphing inequalities offers a visual representation of the solutions. It helps to understand the range of numbers satisfying the inequality. To graph the inequality expressed in interval notation \((-1, \infty)\), follow these steps:

• Start with a number line, a horizontal line with marked numbers.
• Locate \(-1\) on the number line. Since \(-1\) is not part of the solutions (indicated by \(x > -1\)), draw an open circle around it. This circle shows that the value at this point is excluded from the set of solutions.
• From the open circle, draw a line extending to the right. An arrow or line to the right signifies that all numbers greater than \(-1\) are included.
• Extend the line or arrow indefinitely towards positive infinity, reinforcing that there is no upper boundary.

Graphing not only helps solidify understanding but also communicates the solution range in a visual, easily digestible format. This skill is crucial for interpreting inequalities effectively.

Number Line

A number line is a straight, horizontal line that visually represents numbers at equal intervals or distances apart. It's an essential tool in graphing inequalities, helping us display and understand where a particular set of numbers lies in relation to others.

• The center of a number line is usually zero, with positive numbers on the right and negative numbers on the left.
• Each point on the number line corresponds to a real number, providing a spatial idea of its value and relation.
• In graphing inequalities like \( x > -1 \), using a number line becomes invaluable as it shows which numbers are part of the solution set.

When we place an open circle on \(-1\) and extend the line infinitely to the right, we illustrate that the solution includes all numbers greater than \(-1\). A number line supplies a straightforward and visual method to comprehend abstract mathematical concepts such as inequalities.