Problem 33 - Prealgebra and Introductory Algebra

Question

Solve. $$\frac{x}{x+4}=3-\frac{4}{x+4}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -4\).

Step 1: Combine like terms

Combine like terms on each side of the equation. The equation then simplifies to \(\frac{4 + x}{x + 4} = 3\)

Step 2: Cross-multiplication

Cross-multiply which will eliminate the denominator: \(4 + x = 3(x + 4)\). This simplifies as \(x + 4 = 3x + 12\) which can be rewritten as \(2x = -8\).

Step 3: Solve for x

\ To isolate \(x\), divide both sides of the equation by 2 which gives \(x = -4\).

Key Concepts

Fractions

Fractions are an essential part of algebra, representing parts of a whole. They appear as ratios of two numbers: the numerator (top number) divided by the denominator (bottom number). Understanding how to work with fractions enables us to handle various types of equations, especially when variables are involved.
Fractions can sometimes seem daunting, but remember: they are just another way to represent division. For example, if you have the fraction \( \frac{x}{x+4} \), it portrays the division of \(x\) by \(x+4\).
There are basic rules when dealing with fractions, such as:

• Finding a common denominator for addition or subtraction.
• Multiplying fractions directly across numerators and denominators.
• Dividing fractions by multiplying by the reciprocal of the divisor.

In our exercise, being able to combine like terms on each side of the equation was crucial to simplifying the fraction expressions in preparation for solving the equation.

Equations

Equations are mathematical sentences showcasing the equality between two expressions. They are crucial in identifying the value of unknown variables. An essential skill in algebra involves solving equations, which is about finding values for the variables that make the equation true.
Consider the equation form seen in this exercise: \( \frac{x}{x + 4} = 3 - \frac{4}{x + 4} \). The goal is to isolate the variable \(x\) by performing a series of operations.
We usually start by simplifying the equation step-by-step:

• Combine like terms to reduce complexity and form a standard expression.
• Apply mathematical operations to both sides of the equation to maintain balance.
• Isolate the variable, solving for \(x\) or any targeted variable.

In the given problem, combining like terms through proper simplification allowed us to convert the equation into a more manageable form, leading us towards the solution.

Cross-Multiplication

Cross-multiplication is a powerful technique often used to solve equations involving fractions. It involves multiplying each side of the equation in such a way that the denominators are removed, which simplifies the expression significantly.
This process comes into play when you have equations where fractions are set equal to each other or to whole numbers.
To use cross-multiplication effectively, follow these steps:

• Identify the fractions that need to be cross-multiplied.
• Multiply the numerator of one fraction by the denominator of the other and vice versa.
• Set the two resulting expressions equal to each other.

In the step-by-step solution, cross-multiplication was applied to eliminate the denominator \(x+4\) by setting \(4 + x\) equal to \(3(x + 4)\). This process allowed for a clear path to solving for \(x\) by converting the fractional equation into a linear equation. Cross-multiplication simplifies complex fraction problems into direct operations, making it much easier to find the needed variable values.