Problem 75 - Algebra for College Students

Question

1–8, Express the given inequality in interval notation and sketch a graph of the interval. x>1

Step-by-Step Solution

Answer

Interval notation: \((1, \infty)\). Graph: Open circle at 1, shade to the right.

Step 1: Understand the inequality

The inequality given is \(x > 1\). This means that \(x\) is any number greater than 1. There is no upper bound on the value \(x\) can take, which implies \(x\) can be any number up to positive infinity.

Step 2: Determine the interval notation

To express \(x > 1\) in interval notation, we use parenthesis because the number 1 is not included in the solution (since it's a strict inequality). The interval representing all numbers greater than 1 is \((1, \infty)\).

Step 3: Sketch the graph

On a number line, draw an open circle at 1 to indicate that 1 is not included in the interval. Shade the line to the right of 1 extending towards infinity to show that all numbers greater than 1 are included in the interval.

Key Concepts

Inequalities

An inequality is a mathematical statement that indicates the relationship between two expressions. Different symbols are used to represent inequalities, such as:

• \(x > a\) : This means \(x\) is greater than \(a\).
• \(x < a\) : This indicates \(x\) is less than \(a\).
• \(x \ge a\) : Represents \(x\) is greater than or equal to \(a\).
• \(x \le a\) : Means \(x\) is less than or equal to \(a\).

These symbols help describe a range of values, rather than a specific number. In the exercise "\(x > 1\)", it states that \(x\) can be any value larger than 1, but not include 1 itself. Understanding inequalities is important because they are used to express conditions and constraints in various problem-solving scenarios. Consider a simple example from daily life: "I can spend up to \(10." is an inequality stating that my spending should be less than or equal to \)10.

Number Line Representation

Representing inequalities on a number line is a helpful visual tool to understand which numbers satisfy the inequality. For the inequality \(x > 1\), we start by locating the point of interest, that is number 1 on the number line. An important step here is to draw an open circle around number 1. This open circle indicates that \(x = 1\) is not included in the solution because \(x > 1\) is a strict inequality.

Next, shade the number line to the right of the open circle. This shaded region represents all numbers greater than 1, extending towards infinity. Number line representations not only make it easier to visualize the solution set of an inequality, but also provide a clear picture of the relationship exhibited by the inequality. It's a quick way to see which numbers are solutions and which are not.

Interval Graphing

Interval graphing is closely related to inequalities and involves expressing a set of numbers and graphing them. In our exercise, interval notation is used to express where the inequality \(x > 1\) is true. This set of numbers is written as \((1, \infty)\) in interval notation.

Here's what each part means:

• Parentheses \(( )\) are used to denote that the endpoint isn't included. With our example, \((1, \infty)\) tells us that 1 is not part of the interval, as it only includes numbers greater than 1.
• The lower bound of the interval is 1, and the upper bound is represented using infinity (notated as \(\infty\)), since there is no limit to how large \(x\) can be.

Graphically, this interval is represented on the number line by an open circle at 1 and a shaded arrow stretching rightwards. Interval graphing helps quickly identify boundaries and solutions of an inequality.