Problem 50 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(\frac{1}{x}-\frac{2}{2 x+1}=\frac{1}{2 x^{2}+x}\)

Step-by-Step Solution

Answer

The equation is true for all x except x ≠ 0 and x ≠ -1/2.

Step 1: Identify Equivalent Expressions

Recognize that \(2x^2 + x\) is the product \(x(2x+1)\). This means we can replace the denominator on the right-hand side.

Step 2: Get a Common Denominator

The common denominator for all terms is \(x(2x+1)\). Multiply each term by \(x(2x+1)\) to eliminate the denominators.

Step 3: Simplify the Equation

Multiply each term: \[(2x+1) - 2x = 1\]The equation simplifies to \(2x + 1 - 2x = 1\), which further simplifies to \(1 = 1\).

Step 4: Verify the Equation

Since the equation simplifies to the true statement \(1=1\), this indicates that for any \(x\) where the original fractions are defined, the equation holds true. However, check for restrictions where the original fractions are undefined, which are \(x eq 0\) and \(x eq -\frac{1}{2}\).

Key Concepts

Common Denominator

When dealing with equations that involve fractions, finding a common denominator is an essential strategy. Using a common denominator allows us to combine and eliminate fractions, making the equation simpler to solve. In the given problem, we have three different fractional expressions:

• \(\frac{1}{x}\)
• \(\frac{2}{2x+1}\)
• \(\frac{1}{2x^2+x}\)

To combine these fractions, we need to identify a denominator that is common to all. Notice that \(2x^2 + x\) can be broken down into \(x(2x+1)\). This means that \(x(2x+1)\) is the common denominator for all the terms.
By multiplying each term by this common denominator, \(x(2x+1)\), we eliminate the fractions, resulting in a much simpler equation to solve. This step is crucial because it transforms a complex fraction-based problem into a manageable linear equation.

Simplifying Equations

As you solve equations, simplifying them is a step that helps to reveal the solution more clearly. After multiplying by the common denominator, our equation becomes: \[(2x+1) - 2x = 1\]Here, distribute and simplify each side of the equation.
Let's see how we can make it simpler:

• Multiply out terms if necessary.
• Combine like terms.
• Rearrange terms to highlight the solution.

In the example above, the term \(2x + 1 - 2x\) simplifies to \(1\), which naturally leads us to the true statement \(1 = 1\). This shows that the equation is satisfied for all values of \(x\) except those that make the denominator zero.

Equivalent Expressions

Understanding equivalent expressions is key in algebra and particularly useful when simplifying and solving equations. Equivalent expressions are different expressions that represent the same quantity. For instance, in our problem, the expression \(2x^2 + x\) is equivalent to \(x(2x+1)\). This equivalence allows us to re-write the fractions in a way that makes combining and simplifying possible.
Recognizing equivalent expressions helps because it often reveals hidden relationships or simplifies complex terms in an equation. For example:

• They help find common denominators more easily.
• They allow for more straightforward manipulation of algebraic expressions.

By understanding and using equivalent expressions, you can transform and solve equations much more effectively, uncovering solutions and making sense of even seemingly complicated problems.