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Chapter 2: Problem 24

\(x+\frac{2}{3}=-\frac{1}{6}\)

Short Answer

Expert verified
x = -\frac{5}{6}

Step by step solution

01

Isolate the Variable

Subtract \frac{2}{3} from both sides to isolate x. This gives you:\[ x = -\frac{1}{6} - \frac{2}{3} \]
02

Simplify the Right Side

To subtract these fractions, first find a common denominator. The common denominator for 6 and 3 is 6. Rewrite \frac{2}{3} as \frac{4}{6}:\[ x = -\frac{1}{6} - \frac{4}{6} \]
03

Perform the Subtraction

Subtract the fractions:\[ x = -\frac{1}{6} - \frac{4}{6} = -\frac{5}{6} \]

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Key Concepts

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

Isolating the Variable
Isolating the variable means getting the variable by itself on one side of the equation. This helps us find its value. In the equation given, we start with:

\(x + \frac{2}{3} = -\frac{1}{6}\).

We need to subtract \(\frac{2}{3}\) from both sides to isolate \(x\). Subtracting \(\frac{2}{3}\) from both sides looks like this:

\( x = -\frac{1}{6} - \frac{2}{3} \).

Now \(x\) is on its own on the left side of the equation, so we've successfully isolated the variable!
Common Denominator in Fractions
In order to subtract the fractions on the right side, we need them to have the same denominator. A common denominator is a shared multiple of the denominators. For \(\frac{2}{3}\) and \(\frac{1}{6}\), the common denominator is 6.

Here鈥檚 how to make the denominators the same:

  • Identify the least common multiple (LCM) of 3 and 6, which is 6.

  • Rewrite \(\frac{2}{3}\) with the new denominator. Since \(\frac{2}{3} = \frac{2*2}{3*2} = \frac{4}{6}\), we write it as \(\frac{4}{6}\).

Now we have the equation:
\( x = -\frac{1}{6} - \frac{4}{6} \).

We can now subtract the fractions since they have the same denominators.
Fraction Subtraction
When subtracting fractions, keep the denominator the same and subtract the numerators. For our equation, it looks like this:

\(x = -\frac{1}{6} - \frac{4}{6}\).

To subtract:

  • Keep the denominator (6) the same.

  • Combine the numerators: -1 and -4.

This gives us:

\( x = \frac{-1 - 4}{6} = \frac{-5}{6} \).

So, \( x = -\frac{5}{6} \).

Congratulations! We now have our solution.

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