Problem 121 - Elementary Algebra

Question

Find the domain of the equations for the following problems. $$ y=8 x+7 $$

Step-by-Step Solution

Answer

Answer: The domain is all real numbers, represented using interval notation as $$(-\infty, \infty)$$ or using set notation as $$\{x | x \in \mathbb{R}\}$$.

Step 1: Identify the Equation Type

Identify the type of equation given. In this case, the equation is a linear equation: $$y = 8x + 7$$.

Step 2: Check for Restrictions

Check if there are any restrictions or conditions mentioned in the question affecting the domain. In this problem, no restrictions are mentioned.

Step 3: Determine the Domain

Since the equation is a linear equation with no restrictions, the domain includes all real numbers. The domain can be written using interval notation as: $$(-\infty, \infty)$$.

Step 4: Write the Domain in Set Notation

The domain can also be represented using set notation: $$\{x | x \in \mathbb{R}\}$$. This indicates that the domain includes all real numbers x.

Key Concepts

Linear Equations

In mathematics, a linear equation is an equation that creates a straight line when graphed on a coordinate plane. It is characterized by having no variables raised to a power other than one, making it the simplest type of algebraic equation. The standard form of a linear equation in two variables is \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. The slope \( m \) determines how steep the line is, while the intercept \( b \) tells us where the line crosses the y-axis. For the equation \( y = 8x + 7 \), 8 is the slope, and 7 is the y-intercept. This reveals that for every unit increase in \( x \), \( y \) increases by 8 units.

Real Numbers

When we talk about real numbers in mathematics, we are referring to a broad category of numbers that include nearly every number you'll encounter in algebra and everyday life. Real numbers consist of:

• Rational numbers, such as fractions and whole numbers, which can be expressed as a ratio of two integers.
• Irrational numbers, like the square root of 2 or \( \pi \), which cannot be expressed as simple fractions.

The category of real numbers includes both positive and negative numbers, as well as zero. When we state that the domain of a linear equation is all real numbers, it means that we can plug any real number into the equation for \( x \), and we will get a valid \( y \) value.

Interval Notation

Interval notation is a concise way of describing a set of numbers along the number line. It uses parentheses \(()\) and brackets \([]\) to indicate whether endpoints are excluded or included. For instance:

• \((a, b)\) means the interval does not include \( a \) and \( b \) (open interval).
• \([a, b]\) means the interval includes both \( a \) and \( b \) (closed interval).
• \((a, b]\) or \([a, b)\) can include just one of the endpoints.

In the context of the domain of a linear function like \( y = 8x + 7 \), where all real numbers are possible inputs, we use the interval notation \((-\infty, \infty)\). This reflects that there is no minimum or maximum value for \( x \), and it can be any real number, extending infinitely in both directions.

Set Notation

Set notation offers another way to express the domain or range of a function using a mathematical set. In set notation, we often use a description to define the elements in a set. For example, we indicate the domain of a function with the expression \( \{x | x \in \mathbb{R}\} \). Here's what it means:

• The curly braces {} indicate a set.
• The vertical bar \(|\) can be read as "such that."
• \( x \in \mathbb{R} \) specifies that \( x \) is an element of the set of all real numbers.

This notation is crucial when specifying the conditions a variable must meet. In the equation \( y = 8x + 7 \), the domain described in set notation tells us that \( x \) can be any real number. This is another way to convey the same information as interval notation \((-\infty, \infty)\).