Problem 47 - Contemporary Precalculus

Question

Use interval notation to denote the set of all real numbers \(x\) that satisfy the given inequality. $$x \geq-9$$

Step-by-Step Solution

Answer

Question: Represent the inequality, \(x \geq-9\), using interval notation. Answer: \([-9, \infty)\)

Step 1: Understand the inequality

The inequality provided is \(x \geq-9\). This inequality states that \(x\) must be greater than or equal to \(-9\). In other words, any number that is \(-9\) or greater will satisfy this inequality.

Step 2: Use interval notation

To represent the inequality using interval notation, we'll use a closed bracket to indicate that the value \(-9\) is included. The interval will stretch towards positive infinity since there is no upper bound. Thus, the interval notation for this inequality is: $$[-9, \infty)$$

Key Concepts

Inequality

Inequality is a mathematical comparison between two values, indicating that one value is larger or smaller than the other. It provides a way to express a range of numbers that meet a certain condition. For instance, when you see an inequality such as \(x \geq -9\), it’s telling us that \(x\) can be any real number, as long as it is not less than -9.

Inequalities can be strict (with symbols like \(<\) or \(>\)), meaning the values cannot be equal to the number specified, or they can be non-strict (with symbols like \(\leq\) or \(\geq\)), allowing for the value to be equal as well. In the given exercise, the use of \(\geq\) symbol denotes a non-strict inequality, affirming that both -9 itself and any value greater than -9 are solutions to the inequality.

Real Numbers

Real numbers encompass both rational and irrational numbers. They can be thought of as any number that you can find on the number line. This includes integers, fractions, and numbers with endless non-repeating decimals, like \(\pi\).

What's particularly important when discussing inequalities and real numbers is that they describe a continuous range. When you are given the inequality \(x \geq -9\), it doesn’t just mean that integers greater than or equal to -9 are solutions, but rather every possible number, without exception, that is greater than or equal to -9, even those with decimal components, are included.

Closed Bracket

When using interval notation to express ranges of numbers allowed in an inequality, brackets play a significant role in understanding what is included. A closed bracket, denoted by \([\) or \(]\), indicates that a number is part of the interval. It literally 'closes' the interval off and includes the endpoint.

Consider the exercise \(x \geq -9\). In interval notation, this inequality is expressed as \([-9, \infty)\). The closed bracket at -9 conveys that -9 itself is a solution to the inequality, not just the numbers greater than it. This small but crucial detail reflects the precise nature of inequality and is central to accurate mathematical communication.

Positive Infinity

The concept of infinity in mathematics refers to something that is unlimited or without end. Positive infinity, symbolized by \(\infty\), is a way to express that numbers keep increasing indefinitely. It's more of an idea than an actual number you could ever reach.

In the context of inequalities, when an interval extends to positive infinity, like in \([-9, \infty)\), it means there is no upper limit to the values that \(x\) can take. The round bracket next to infinity indicates it is not a number we can include, since infinity itself cannot be reached or fixed. It’s a way to express that the set of numbers continues forever in the positive direction.