Problem 86 - Elementary Algebra

Question

Solve each of the following equations. Don't forget that division by zero is undefined. $$\frac{x+3}{x}=1+\frac{3}{x}$$

Step-by-Step Solution

Answer

The solution is all values of \( x \neq 0 \).

Step 1: Identify the Equation

The equation given is \( \frac{x+3}{x} = 1 + \frac{3}{x} \). We are to find values of \( x \) where this holds true and remember that \( x eq 0 \) because division by zero is undefined.

Step 2: Simplify Both Sides

Notice that the left side of the equation is already simplified: \( \frac{x+3}{x} \). We rewrite the right-hand side as a single fraction: \( 1 + \frac{3}{x} = \frac{x}{x} + \frac{3}{x} = \frac{x+3}{x} \).

Step 3: Compare Both Sides of the Equation

After rewriting, the equation is \( \frac{x+3}{x} = \frac{x+3}{x} \). By comparing these, we see that both sides are equal for all \( x eq 0 \).

Step 4: Identify Values of x

Since both expressions \( \frac{x+3}{x} \) and \( \frac{x+3}{x} \) are identical for \( x eq 0 \), it confirms that every \( x eq 0 \) is a solution.

Key Concepts

Division by Zero

When solving algebra equations, one important rule we must remember is that division by zero is undefined. In simple terms, dividing any number by zero does not produce a valid result. This is because division asks the question: "How many times does the divisor fit into the dividend?" When the divisor is zero, this question cannot be answered satisfactorily because zero cannot be multiplied by any number to productively reach another number.

To avoid errors in calculations, always check the denominator. If there is any variable in your denominator, like the given equation \( \frac{x+3}{x} = 1 + \frac{3}{x} \), ensure that the value of that variable does not create zero as a divisor. In this case, \( x eq 0 \). Keep this in mind as you encounter fractions in algebra; never allow a situation where you're dividing by zero.

Simplifying Fractions

Simplifying fractions is a crucial step in solving algebraic equations. It involves reducing fractions to their simplest form by performing operations that combine like terms or factor common elements.

For the equation given: \[ \frac{x+3}{x} = 1 + \frac{3}{x} \] we start by simplifying the right side. Notice how we converted \[ 1 + \frac{3}{x} \] into a single fraction:

• Transform 1 into a fraction with a denominator matching the other fraction: \( \frac{x}{x} \).
• Combine into a single expression: \( \frac{x}{x} + \frac{3}{x} = \frac{x+3}{x} \)

Simplifying like this aligns both sides of the equation, making it easier to see if they are equivalent or to perform further operations. It's a fundamental skill for manipulating expressions and equations efficiently.

Equivalent Equations

Determining if equations are equivalent is crucial in algebra as it allows us to confirm solutions or make decisions about the variable values. Two expressions or equations are equivalent if they simplify to the same form under the same conditions.

In the given problem, after simplifying both sides, we reached the point where both sides of the equation look identical:\[ \frac{x+3}{x} = \frac{x+3}{x} \] This result shows the expressions themselves are equivalent, meaning they will hold true for the same values of \( x \), provided the initial condition \( x eq 0 \) is respected due to the division by zero issue.

Understanding equivalent equations can greatly help in verifying your work when solving equations or discovering new solutions, ensuring your approach to the problem is correct and logical.