Question

Simplify the expression. $$\frac{4 x}{5 x-2}-\frac{2 x}{5 x+1}$$

Short answer

The simplified form of the given expression is \( \frac{2(5x + 4)}{(5x - 2)(5x + 1)} \)

Step-by-step solution

Finding the Common Denominator

The common denominator LCD of the two fractions \( \frac{4x}{5x-2} \) and \( \frac{2x}{5x+1} \) would be the product of their individual denominators, which is \((5x - 2)(5x + 1)\).

Combining the Fractions

The given expression can now be rewritten as follows: \( \frac{4x(5x+1)}{(5x - 2)(5x + 1)} - \frac{2x(5x - 2)}{(5x - 2)(5x + 1)} \)

Simplifying the Numerators

Distribute each term in each numerator to get \( \frac{20x^2 + 4x}{(5x - 2)(5x + 1)} - \frac{10x^2 - 4x}{(5x - 2)(5x + 1)} \), and then simplify the numerator to get \( \frac{20x^2 + 4x - 10x^2 + 4x}{(5x - 2)(5x + 1)} \), it simplifies to \( \frac{10x^2+8x}{(5x - 2)(5x + 1)} \)

Further Simplification

The numerator \( 10x^2+8x \) can be factored out common term \( 2x \) and we'll get \( \frac{2x(5x + 4)}{(5x - 2)(5x + 1)} \). Finally, we can divide both the numerator and the denominator by \( x \) which yields \( \frac{2(5x + 4)}{(5x - 2)(5x + 1)} \), assuming \( x \neq 0 \).

Key concepts

Common Denominator

Understanding the concept of a common denominator is crucial when working with fractions. It's the key to combining fractions in such a way that you can easily add, subtract, or compare them. The common denominator refers to a shared multiple of the denominators of two or more fractions. To find it, simply take the denominators of the fractions you wish to combine and identify the least common multiple (LCM) of these numbers. In our exercise, for instance, the denominators are unique expressions, so the common denominator is the product of these expressions. It's essential to remember that we seek the least complex form that is common to all denominators to simplify the computations.

Combining Fractions

The process of combining fractions involves adding or subtracting them, which can only be done efficiently when the fractions share a common denominator. Once we have identified the common denominator, we rewrite each fraction so that they have this denominator, and then simply combine the numerators keeping the common denominator intact. In the exercise, after the fractions are rewritten with the common denominator, the numerators are combined by adding or subtracting them as indicated. Keep in mind that while the denominators remain the same, the numerators undergo the algebraic operations.

Factoring Algebraic Expressions

Factoring is a powerful tool in algebra that allows us to simplify expressions and solve equations. When we factor algebraic expressions, we are essentially breaking them down into the product of simpler expressions. Think of it like reverse multiplication. For example, in the expression from our exercise, the numerator can be factored by identifying the greatest common factor (GCF) which, in this case, is a variable term. Pulling out the GCF simplifies the expression, making it easier to reduce fractions or solve equations. Factoring is often used alongside other algebraic techniques, such as expanding expressions using the distributive property, to fully simplify an expression.

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term within a parenthesis. The form is typically \( a(b + c) = ab + ac \). It is particularly useful when we have to simplify complex expressions as we do with the numerators in our example. This property ensures that multiplication is properly distributed over addition or subtraction inside the parentheses. When simplifying algebraic expressions, such as in the step-by-step solution, applying the distributive property correctly is essential to reach a more manageable form of expression before further simplification steps, like factoring, can be applied.