91Ó°ÊÓ

When it comes to solving equations, the objective is to find the value of the unknown variable that makes the equation true. In our exercise example, the equation given is \(x + 5 = 7\). To solve this, one can use inverse operations that 'undo' the effect of a previous operation. Here, the inverse operation is subtraction because the original equation involves addition.

To isolate the variable 'x', we subtract 5 from both sides of the equation resulting in \(x = 2\). This effectively 'cancels out' the addition of 5 on the left side, finding the value for 'x' that satisfies the equation. This process is fundamental in algebra and is the first step towards understanding more complex mathematical concepts.

• Identify the operation that has been applied to the variable.
• Apply the inverse operation to both sides of the equation.
• Simplify the equation to solve for the variable.

In the context of elementary set theory, a set is a collection of distinct objects, considered as an object in its own right. Sets can be described in various ways, one of which is the roster method, as seen in our exercise. The roster method lists all elements of a set within curly brackets, with each element separated by a comma.

For example, if a set contains the numbers 1, 2, and 3, using the roster method, it would be expressed as \(\{1, 2, 3\}\). The given exercise asks to express the set of values that satisfy the equation \(x+5=7\) using this method. Since the solution to the equation is \(x=2\), the set contains just one element and is written as \(\{2\}\).

• A set is a collection of distinct elements.
• The roster method lists all elements explicitly within curly brackets.
• Each element in the roster is unique and separated by commas.

Understanding mathematical notation is crucial for communicating and comprehending mathematical ideas clearly and compactly. Notation is a system of symbols and signs used to represent numbers, operations, relations, and other concepts within mathematics.

In the example, \(\{x \: x+5=7\}\) combines notation for sets and equations. The curly brackets represent a set, and the vertical bar \(\textbar\) is read as 'such that' signifying a condition for elements in the set. The equation \(x+5=7\) within the set notation serves as a condition that defines the elements of the set. When such notations are used appropriately, they can significantly streamline the process of understanding and solving mathematical problems.

• Curly brackets \(\{ \}\) denote a set in mathematics.
• The vertical bar \(\textbar\) is used for specifying a condition in set-builder notation.
• Equations within set notation imply that elements must satisfy the equation to be part of the set.