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Solving equations is a fundamental part of algebra that involves finding the values of variables that make the equation true. An equation is like a balance scale. On both sides of the equal sign, you have amounts that should be equal when balanced. In our example, the equation is initially:

• \(4(3x + 2) = 12x + 8\)

The goal is to find the value(s) of \(x\) that will make both sides equal.

It's important to note that solving equations isn’t just about moving terms to simplify; it’s about understanding what makes an equation true. Sometimes, as shown in this example, upon simplification, the left and right sides of our equation are identical, meaning the equation is always satisfied regardless of the value of \(x\). This brings us to the result: \(x\) can be any real number, illustrating an identity in algebra.

The distributive property is a crucial algebraic principle where you multiply a single term by each term inside a parenthesis. It is expressed generally as \(a(b + c) = ab + ac\).

• In our equation, \(4(3x + 2)\), the distributive property tells us to multiply \(4\) by both \(3x\) and \(2\).
• This results in \(12x + 8\).

By applying the distributive property, you simplify expressions and make them easier to work with when solving equations. It's like breaking down a complex problem into smaller, more manageable parts. This property helps maintain balance in the equation, meaning the value stays the same despite the transformation. It’s a fundamental skill that you'll use often in algebra to manipulate and solve equations effectively.

Real numbers are all the numbers you can think of on the number line, including both rational numbers, like fractions and whole numbers, and irrational numbers, like the square root of 2 and \(\pi\). When solving equations, knowing whether solutions belong to the set of real numbers is crucial.In our exercise, we discovered that \(x\) could be any real number. This means no matter what real number you substitute for \(x\), the equation remains true:

• This occurs because both sides of our equation are actually the same.
• In simplified form, this means the left equals the right regardless of the value for \(x\).

This property is known in algebra as an identity, meaning it holds for all possible inputs within the realm of real numbers. When thinking about real numbers, imagine the freedom to use any whole number, fraction or decimal; this empowers the equation to hold true consistently across all these inputs.