Problem 54 - Precalculus Mathematics for Calculus

Question

Express the inequality in interval notation, and then graph the corresponding interval. $$x \geq-5$$

Step-by-Step Solution

Answer

The inequality \( x \geq -5 \) is written as \([-5, \infty)\) in interval notation and is graphed as a solid dot at \(-5\) with a line extending to the right.

Step 1: Understand the Inequality

The inequality given is \( x \geq -5 \). This means that \( x \) can be any number greater than or equal to \(-5\).

Step 2: Convert Inequality to Interval Notation

In interval notation, \( x \geq -5 \) is written as \([-5, \infty)\). The bracket \([\) indicates that \(-5\) is included in the interval, and the parenthesis \()\) indicates that infinity is not included.

Step 3: Graph the Interval on a Number Line

To graph \([-5, \infty)\) on a number line, draw a solid dot or filled circle at \(-5\) to show that \(-5\) is included in the interval. Draw a line extending to the right from \(-5\) to represent all numbers greater than \(-5\). Since \(\infty\) is not an actual number, the line extends infinitely to the right.

Key Concepts

Interval Notation

Interval notation is a shorthand way to represent ranges of numbers, particularly useful for expressing solutions to inequalities. It uses symbols, like brackets and parentheses, to show which numbers are included in a set:

• A square bracket \(\[ \]\) is used to indicate a closed interval, meaning the endpoint is included. For example, \([-5, 2]\) includes all numbers between and including \(-5\) and \(2\).
• A parenthesis \(\( \)\) indicates an open interval, where the endpoint is not included. For instance, \([2, \infty)\) includes all numbers greater than \(2\) but not \(2\) itself.

For the inequality \(x \geq -5\), the interval notation is \([-5, \infty)\). This tells us that \(-5\) is part of the solution because of the square bracket, while \(\infty\), being symbolic for a concept that has no end, is never included.

Number Line

A number line visually represents real numbers as points on a straight line. Each point corresponds to a unique number, making it a simple and effective tool for visualizing intervals and understanding inequalities.

• To use a number line effectively, first mark the points of interest, such as endpoint numbers, using dots or circles.
• Solid dots indicate numbers that are included in an interval (closed endpoints), while open circles show numbers that are not included (open endpoints).

For the interval \([-5, \infty)\), start by placing a solid dot at \(-5\) on the number line, reflecting that this endpoint belongs to the interval. Then, extend a continuous line to the right, representing all numbers greater than \(-5\) extending infinitely.

Graphing Inequalities

Graphing inequalities involves representing solutions visually, usually on a number line or coordinate plane, showing a range of possible values that satisfy the inequality.
When graphing on a number line, follow these steps:

• Identify the type of inequality: "less than" (<), "greater than" (>), "less than or equal to" (≤), or "greater than or equal to" (≥).
• Mark key points: Utilize a solid dot for "equal to" and an open circle for "not equal to" conditions, indicating whether specific numbers are included in the solution.
• Shade portions of the line: Extend a line or shade the region to show all solutions. Use an arrow indicating the direction of inequality (left for less than, right for greater than) and ensure it reflects any infinite extension with an open arrowhead.

For \(x \geq -5\), draw a solid dot at \(-5\), then shade or extend the line to the right, representing the inclusion of every number greater than \(-5\), reaching towards infinity.