Problem 73 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$\frac{x}{2}-\frac{x}{4}+4=x+4$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{x}{2}-\frac{x}{4}+4=x+4\) is x=0.

Step 1: Simplify the Equation

First, try to subtract \(\frac{x}{4}\) from \(\frac{x}{2}\) by finding a common denominator. The common denominator of 2 and 4 is 4. Convert \(\frac{x}{2}\) to \(\frac{2x}{4}\), so the equation becomes \(\frac{2x}{4}-\frac{x}{4}+4=x+4\). Now subtract the fractions to get \(\frac{x}{4}+4=x+4\).

Step 2: Solve for x

Now, isolate x on one side of the equation. Begin by multiplying the entire equation by 4 to clear out the fraction. This would give \(x+16=4x+16\). Subtract x and 16 from both sides to get \(0=3x\). Divide by 3 to get \(x=0\).

Step 3: Check the Solution

Finally, substitute the found value, x=0, back into the original equation to check if it holds. \(\frac{0}{2}-\frac{0}{4}+4=0+4\). This simplifies to 4=4, this proves that the solution is correct.

Key Concepts

Algebraic Fractions

Understanding algebraic fractions is essential when solving equations that include fractions with variables. An algebraic fraction is similar to a regular fraction, but instead of numerical denominators and numerators, it incorporates algebraic expressions.

For instance, in the equation \( \frac{x}{2}-\frac{x}{4}+4=x+4 \), both terms \( \frac{x}{2} \) and \( \frac{x}{4} \) are examples of algebraic fractions. Here's the trick to simplify these: find a common denominator. The common denominator makes it easier to combine or cancel out these fractions. We usually look for the smallest number that both denominators divide into, known as the Least Common Denominator (LCD). In this case, the LCD is 4.

Identifying Like Terms

After finding a common denominator, the next step is to convert each fraction to have this common denominator. Sometimes, this means multiplying the numerator and the denominator to create equivalent fractions. Once converted, we can combine like terms, which are terms with the same variable raised to the same power. In our example, \( \frac{x}{2} \) becomes \( \frac{2x}{4} \), so that it has the same denominator as \( \frac{x}{4} \), making it possible to subtract these terms.

Equation Solving Step by Step

When approaching the challenge of solving an equation, following a step-by-step process can make the task manageable. Let's break down these steps using our exercise as a model:

1. Simplify the Equation: Combine like terms and use common denominators for fractions, as done by converting \( \frac{x}{2} \) to \( \frac{2x}{4} \) and subtracting the fractions.
2. Isolate the Variable: Move terms to get the unknown variable on one side of the equation. Here, we multiplied by 4 to get rid of the fraction and moved all x terms to one side.
3. Check the Solution: Always substitute the found value back into the original equation to ensure it satisfies the equation. Here, when we check x=0, both sides of the original equation equal 4, reaffirming our solution.

This structured approach to equation solving helps ensure each operation is performed carefully, and nothing is overlooked. Clearing fractions and combining terms may appear straightforward, but paying close attention to these steps is crucial for correctly solving equations.

No Solution and Identity Equations

There are special kinds of equations to be aware of: no solution equations and identity equations. An equation has no solution when no value of the variable can make the equation true. This typically leads to a statement that is logically false, such as 0=3.

As for identity equations, they are true for all values of the variable, leading to a tautology like 4=4. Identifying these types of equations is part of solving them. If simplifying an equation results in a false statement, it has no solution. But if it simplifies to a true statement, regard the initial equation as an identity.

The concept applies to our example in a subtle way. During the solution process, we arrive at a statement 4=4 after substituting x=0. This confirms that x=0 is the solution. However, had we arrived at a contradiction, we would have deemed the equation unsolvable, and had it been true for any x (like x−x=0), then it would be an identity.