Problem 24 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{4}{x+2}=1 $$

Step-by-Step Solution

Answer

Answer: x = 2

Step 1: Clear the denominator

To clear the denominator, x+2, we will multiply both sides of the equation by x+2. \((x+2)\cdot \frac{4}{x+2} = 1 \cdot (x+2)\)

Step 2: Simplify the equation

The (x+2) terms on the left side cancel each other out, leaving us with an equation without fractions: \(4 = x+2\)

Step 3: Solve for x

In order to find the value of x, subtract 2 from both sides of the equation: \(4-2 = x+2-2\) \(2 = x\) So, x = 2 is the solution to the given rational equation: $$ \frac{4}{x+2} = 1 $$

Key Concepts

Clearing Denominators

When it comes to solving rational equations like \( \frac{4}{x+2}=1 \), the first challenge you might encounter is the presence of a denominator. In elementary algebra, a common technique to tackle this is known as clearing the denominators. This means you want to get rid of any fractions by getting a common denominator or, if possible, eliminating the denominator entirely.

To clear the denominator, you can multiply both sides of the equation by the denominator itself. In this case, multiplying both sides by \( x+2 \) will remove the fraction, because the denominator on the left side and the multiplying factor cancel each other out. This step makes your equation easier to manage and sets the stage for further simplification. Just remember that when you multiply one side of the equation, you must do the same to the other side to maintain balance.

Simplifying Equations

After you've cleared the denominator in a rational equation, the next phase is simplifying the equation. Simplifying can encompass various steps but typically involves combining like terms, canceling out identical terms on both sides, and reducing expressions to their simplest form.

In our example, after clearing the denominator, we obtain \( 4 = x+2 \) with no denominators in sight. Now, the equation looks like something you would see in an early algebra class. It's critical to simplify correctly because it makes identifying the solution much more straightforward. Always double-check your simplification to ensure accuracy; even small mistakes during this stage can lead to incorrect results.

Elementary Algebra

With the simplified equation in hand, we arrive at an important branch of mathematics: elementary algebra. This area of study is concerned with solving equations for an unknown variable. The equation \( 4 = x+2 \) is a basic example of an algebraic equation where our goal is to solve for \( x \).

Elementary algebra uses a set of rules called 'properties of equality' to solve equations. These include actions like adding, subtracting, multiplying, or dividing both sides of the equation by the same number. These rules serve to isolate the variable on one side of the equation, which is precisely what we aim to achieve in our current problem. Understanding these rules is essential as they form the foundation of more complex algebraic concepts that students will encounter in their studies.

Rational Expression Solution

The final step in solving a rational equation is finding the rational expression solution. A rational expression is simply the quotient of two polynomials. Our initial problem \( \frac{4}{x+2}=1 \) is an example of a rational expression set equal to a number. Once we've cleared the denominators and simplified the equation, we can solve for the variable, which represents our unknown value.

In our example, we find that \( x = 2 \) after simplifying and applying elementary algebra. This means that when \( x \) is substituted back into the original rational expression, it should make the equation true. It's vital to check your solution by plugging it back into the original equation to ensure that it's valid and doesn't result in an undefined expression, such as division by zero.