Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Show that the equation is an identity. $$\frac{3 x^{2}+8}{x}=\frac{8}{x}+3 x$$

Step-by-Step Solution

Answer

The equation is an identity because both sides simplify to the same expression, \( 3x + \frac{8}{x} \).

Step 1: Identify the Two Sides of the Equation

The given equation is \( \frac{3x^2 + 8}{x} = \frac{8}{x} + 3x \). We need to show that the left-hand side (LHS) is equal to the right-hand side (RHS).

Step 2: Simplify the Left-Hand Side (LHS)

Start by simplifying the LHS: \( \frac{3x^2 + 8}{x} \). Distribute the division by \( x \) across the numerator to get: \[ \frac{3x^2}{x} + \frac{8}{x} = 3x + \frac{8}{x}. \]

Step 3: Simplify the Right-Hand Side (RHS)

The RHS of the equation is already given as \( \frac{8}{x} + 3x \). Observe that this is already in the simplest form.

Step 4: Compare LHS and RHS

Now compare the expressions obtained from the LHS and the RHS. The simplified form of the LHS is \( 3x + \frac{8}{x} \), which is identical to the RHS, \( \frac{8}{x} + 3x \).

Step 5: Conclusion

Since the LHS simplifies to the same expression as the RHS, \( 3x + \frac{8}{x} \), this means the original equation is an identity. This shows that no matter the value of \( x \), provided \( x eq 0 \), both sides will always equal the same expression.

Key Concepts

Identities in Algebra

In algebra, an identity is a statement that holds true for all values of the variables within a certain range. This means that both sides of the equation are the same for any value that makes the expression valid, typically all real numbers except those that lead to undefined situations. For instance, in the example given, \( \frac{3 x^{2}+8}{x} = \frac{8}{x}+3 x \), we can say this equation is an identity because after simplification, both sides match perfectly.

Identities are helpful as they indicate a fundamental sameness between expressions, offering a reliable tool for reorganizing and simplifying expressions in algebra. Recognizing identities allows us to prove that two expressions are equivalent without evaluating them for every possible value. It is akin to discovering a universal truth within a mathematical framework.

When working with identities, remember:

• They hold true for all permissible values of the variable.
• Simplifying both sides to check their equivalency is essential.
• If left-side and right-side transformations result in the same expression, it indicates an identity.
• Observing any restrictions like division by zero is crucial to ensure the identity remains valid.

Simplifying Expressions

Simplifying algebraic expressions involves performing operations to rewrite the expression in its simplest form, making it easier to work with. In this exercise, the simplification process involves dividing each term in the numerator by the common denominator \( x \).

For the expression \( \frac{3x^2 + 8}{x} \):

• First, perform the division separately: \( \frac{3x^2}{x} + \frac{8}{x} \)
• The first term simplifies to \( 3x \), and the second term remains \( \frac{8}{x} \)
• Thus, the simplified expression becomes \( 3x + \frac{8}{x} \).

This approach helps ensure that complex expressions are broken down into their simpler components, easing the process of comparing them or performing further operations. Always aim to reduce expressions fully, as it provides clarity and enhances understanding.

Key points to keep in mind when simplifying include:

• Combine like terms if possible.
• Always watch for opportunities to factor expressions or cancel common factors.
• Avoid introducing extraneous terms or errors during simplification.

Simplification is often the crucial step for solving equations or proving identities, as it clarifies the mathematical relationships.

Equation Solving

Solving equations in algebra involves finding the values for variables that make the equation true. In cases like identities, solving means demonstrating the equality of expressions for all variable values, as long as those values are within the equation's defined domain.

When presented with an equation like \( \frac{3x^2 + 8}{x} = \frac{8}{x} + 3x \), the goal is to simplify or manipulate both sides to determine if they are indeed identical. This involves:

• Breaking down complex expressions for clarity.
• Performing operations consistently on both sides of the equation.
• Checking the equivalence through simplification, as in this case, resulting in \( 3x + \frac{8}{x} = \frac{8}{x} + 3x \).

Rather than solving for a specific value, indicating all permissible values prove the identity. It's essential to note any restrictions such as \( x eq 0 \), which prevents the equation from being undefined.

While idiosyncratic solving relies primarily on understanding and manipulating the structure of the equation, always ensure steps are transparent and logical. Tracking how transformations maintain the equation's integrity is vital for accurate problem-solving.