Question

Identify the least common denominator of the equation. $$\frac{4}{x+9}+\frac{3 x}{2 x-1}=\frac{10}{3}$$

Short answer

The least common denominator for the given equation is \((x + 9) (2x - 1) * 3\).

Step-by-step solution

Identify the Denominators

Here, the denominators in the equation are \(x+9\), \(2x-1\), and 3. These are found at the bottom of each fraction in the equation.

Find the Least Common Denominator

A common denominator of the terms can be obtained by multiplying all the individual denominators together. So, the least common denominator for this equation will be \((x + 9) (2x - 1) * 3\).

Key concepts

Understanding Rational Expressions

Rational expressions are the algebraic equivalent to fractions, where both the numerator and the denominator can contain variables. Much like fractions that contain only numbers, the denominator in a rational expression cannot equal zero because division by zero is undefined. An example of a rational expression is \(\frac{4}{x+9}\), where \(x+9\) must not be zero. Finding a least common denominator (LCD) is crucial when you need to add, subtract, or compare rational expressions with different denominators, just as you would for numerical fractions.

Demystifying Algebraic Fractions

Algebraic fractions behave just like regular fractions but include variables. The denominator represents the number of equal parts the whole is divided into, and the numerator represents how many parts are being considered. Understanding this concept allows us to solve for unknown variables within these fractions. If we consider our example, we have \(\frac{3x}{2x-1}\), which signifies a certain multiple of \(x\) divided into \(2x-1\) parts. Working with algebraic fractions often requires finding a least common denominator (LCD), which helps in simplifying and solving equations involving multiple fractions.

Solving Rational Equations with Ease

Solving rational equations often starts by finding the LCD to eliminate the fractions. In our exercise, the LCD for \(\frac{4}{x+9}\), \(\frac{3x}{2x-1}\), and \(\frac{10}{3}\) is their product \((x+9)(2x-1)*3\). Once we identify this LCD, we multiply each term in the equation by it. This step helps to clear the fractions, leaving a simpler equation without denominators that can be solved using standard algebraic methods. It's essential to always check your solutions in the original equation to avoid extraneous solutions that could potentially make any denominators equal zero.