Question

Identify the least common denominator of the equation. $$\frac{x}{x+3}+\frac{1}{x}=\frac{3}{x}$$

Short answer

The least common denominator for the equation is \$x(x+3)\$.

Step-by-step solution

Identify the denominators

In this equation, the denominators are identified as being \$x+3\$, \$x\$, and \$x\$ respectively.

Analyze the denominators

In order to find the least common denominator, it should be first determined if there is a factor common in all these denominators. Recognize, that in this case, two fractions share \$x\$ as a denominator.

Find the least common denominator

Finally, the least common denominator would include all the unique factors from the denominators. As observed, the denominators are infinite values corresponding to either \$x\$ or \$x+3\$. Their least common denominator is simply the product of these two denominators, which is \$(x)(x+3)\$ or \$x(x+3)\$.

Key concepts

Algebraic Fractions

Algebraic fractions are just like regular fractions, but instead of numbers, they have algebraic expressions in the numerator, the denominator, or both. In the given problem, we have fractions like \(\frac{x}{x+3}\) and \(\frac{1}{x}\), where the numerators and denominators involve variables. Handling algebraic fractions generally involves finding a common denominator so that you can combine them or compare them effectively.

When dealing with algebraic fractions, it’s essential to remember that the rules of arithmetic fractions still apply. We can add, subtract, multiply, and divide them using the same principles, but with the added complexity of the algebraic expressions.

Equation Solving

Equation solving is a fundamental skill in algebra that involves finding the values of unknown variables that make the equation true. In this problem, we're aiming to solve for \(x\) by first finding a common denominator for all the algebraic fractions involved. Once a common denominator is found, the algebraic fractions can be rewritten, combined, and simplified to make the equation more manageable.

Effective equation solving requires not only algebraic manipulation but also an understanding of how to deal with the different forms of algebraic expressions, especially when fractions are involved. Isolating the variable on one side to solve for it thereafter is a key step in this process.

Common Denominator

Finding a common denominator is crucial when working with fractions, particularly when they need to be added or compared. It means identifying a shared multiple of all the denominators involved. In simple numerical fractions, we usually look for the smallest number that each denominator can divide into. With algebraic fractions, it's about finding an expression that incorporates all the factors from the original denominators without repetition.

In our textbook exercise, the common denominators are variable expressions. Identifying the least common denominator (LCD) simplifies the process of combining the fractions. The LCD for our exercise is \(x(x+3)\), as this expression can be divided by each of the individual denominators of our fractions. This understanding is essential to correctly combine and simplify the fractions involved.

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator and the denominator, much like the algebraic fractions we are working with. When dealing with rational expressions, it's important to determine their domain, which includes all the values the variables can take without making the denominator zero — a crucial first step before performing operations with them.

Operations like addition, subtraction, multiplication, and division are performed on rational expressions using the same basic principles as with numerical fractions, but one must also consider the implications of variable expressions and their allowable values. Handling rational expressions adeptly is key to mastering algebra and progressing in mathematics.