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Chapter 4: Problem 45

Solve the equation and simplify your answer. $$x-\frac{1}{4}=-\frac{1}{8}$$

Short Answer

Expert verified
The solution to the equation is \(x = \frac{1}{8}\).

Step by step solution

01

Isolate the variable x

To isolate the variable \(x\), we need to add \(\frac{1}{4}\) to both sides of the equation:\[x - \frac{1}{4} + \frac{1}{4} = -\frac{1}{8} + \frac{1}{4}\] This simplifies to:\[x = -\frac{1}{8} + \frac{2}{8}\]
02

Perform the addition

Now, perform the addition on the right side. The denominators are the same, so we can simply add the numerators:\[-\frac{1}{8} + \frac{2}{8} = \frac{2-1}{8} = \frac{1}{8}\] The simplified equation becomes:\[x = \frac{1}{8}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractions
Fractions are a way to express numbers as parts of a whole. They consist of a numerator, which is the top part, and a denominator, which is the bottom part. Understanding fractions is crucial when solving equations, particularly when they are included in the equation. In the given exercise, we encounter fractions like \(-\frac{1}{8}\) and \(\frac{1}{4}\).

Here鈥檚 a quick breakdown of how they work:
  • The numerator indicates how many parts we have.
  • The denominator shows into how many parts the whole is divided.
With fractions, keep in mind that you need a common denominator to add or subtract them efficiently. For instance, both \(-\frac{1}{8}\) and \(\frac{1}{4}\) are dealt with over the common denominator of 8 before they can be combined.
Isolate Variable
Isolating the variable means getting the variable by itself on one side of the equation. This step is essential for finding the solution to the equation. In the exercise, the variable to solve for is \(x\).

To isolate \(x\), we need to move all other terms to the opposite side. We achieve this by performing the same operation on both sides of the equation, which keeps the equation balanced.

In our example:
  • We start with \(x - \frac{1}{4} = -\frac{1}{8}\).
  • We add \(\frac{1}{4}\) to both sides, which results in \(x = -\frac{1}{8} + \frac{1}{4}\).
We now have \(x\) isolated, allowing us to simplify the right side to find its value.
Simplify Expressions
Simplifying expressions involves performing all possible arithmetic operations to express the equation in its simplest form. After isolating the variable, our task is to simplify what remains.

For the equation we are solving, \(x = -\frac{1}{8} + \frac{1}{4}\), we need to perform the addition of the fractions. It requires a common denominator:
  • The expressions \(-\frac{1}{8}\) and \(\frac{1}{4}\) have been converted to the same denominator of 8.
  • This allows us to simplify: \(-\frac{1}{8} + \frac{2}{8} = \frac{1}{8}\).
Therefore, the final simplified expression gives us the solution \(x = \frac{1}{8}\), making the variable clear and straightforward to interpret.

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Most popular questions from this chapter

Is \(1 / 4\) a solution of the equation \(x+\frac{1}{3}=\frac{5}{12}\) ?

Solve the equation and simplify your answer. $$-\frac{3}{2} x+\frac{8}{3}=\frac{7}{9} x-\frac{1}{2}$$

Simplify the expression. $$-\frac{1}{3}-\frac{1}{5}\left(-\frac{4}{3}\right)$$

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