Question

Simplify the expression. $$\frac{3 x+10}{7 x-4}-\frac{x}{4 x+3}$$

Short answer

The simplified form of the given expression is \(\frac{5x^2 + 74x + 30}{(4x + 3)(7x - 4)}\).

Step-by-step solution

Finding the Common Denominator

To subtract two fractions, we first need to have a common denominator. In this case, the denominators are \(7x - 4\) and \(4x + 3\). The common denominator of these two fractions would be the product of these two denominators. Hence, the common denominator will be \((7x - 4)(4x + 3)\).

Redefining the Fractions

Now, redefine the fractions using the common denominator we just found. The first fraction: \(\frac{(3x + 10)(4x + 3)}{(7x - 4)(4x + 3)}\) and the second fraction: \(\frac{x(7x - 4)}{(4x + 3)(7x - 4)}\). Subtracting the second fraction from the first we get the final expression: \(\frac{(3x + 10)(4x + 3) - x(7x - 4)}{(4x + 3)(7x - 4)}\).

Simplifying the Expression

Expand the numerators of the fractions to break down the brackets. You will get: \(\frac{12x^2 + 30x + 40x + 30 - 7x^2 + 4x}{(4x + 3)(7x - 4)}\). Simplify the expression in the numerator by collecting and simplifying like terms: \(\frac{5x^2 + 74x + 30}{(4x + 3)(7x - 4)}\). This is the simplest form of the given expression.