Problem 30 - Algebra 2

Question

Solve each equation. Check your solutions. \(|6 c-1|=0\)

Step-by-Step Solution

Answer

The solution is \(c = \frac{1}{6}\).

Step 1: Understand the Absolute Value Expression

We have the equation \(|6c - 1| = 0\). Recall that the absolute value of a number is zero only if the number itself is zero. Therefore, we need to solve the equation inside the absolute value for zero: \(6c - 1 = 0\).

Step 2: Solve the Equation

Now solve the equation \(6c - 1 = 0\). Add 1 to both sides to isolate the term with \(c\): \[6c - 1 + 1 = 0 + 1\]This simplifies to \(6c = 1\).

Step 3: Isolate the Variable

Divide both sides by 6 to solve for \(c\): \[c = \frac{1}{6}\]So, \(c\) is equal to \(\frac{1}{6}\).

Step 4: Check the Solution

Substitute \(c = \frac{1}{6}\) back into the original equation to verify: \[|6 \left(\frac{1}{6}\right) - 1| = |1 - 1| = |0| = 0\]Since the left side equals the right side, \(c = \frac{1}{6}\) is indeed a solution.

Key Concepts

Equation Solving

When we encounter an equation, like \(\left|6c - 1\right| = 0\), we need to find the value of the variable \(c\) that makes the equation true. Absolute value equations require special attention: they express that the number inside the absolute value brackets \(\left|x\right|\) is equal to zero, so the expression itself must also be zero.

• Recognize that an absolute value can only be zero if the number inside it is zero.
• Identify the expression inside the absolute value brackets.
• In this case, solve \(6c - 1 = 0\).

Taking these steps leads you to focus on the expression within the absolute value and convert the problem to simpler terms, setting you up for success in solving the equation.

Variable Isolation

Isolation of a variable is crucial in solving equations effectively. Our goal is to get \(c\) by itself on one side of the equation, untouched by other numbers or variables. For the equation \(6c - 1 = 0\), do the following:

• Add 1 to both sides to simplify:
• \[6c - 1 + 1 = 0 + 1\]
• This clarifies to \(6c = 1\).
• Next, divide both sides by 6 to isolate \(c\): \[c = \frac{1}{6}\]

This approach involves performing inverse operations to "clear away" any numbers or terms attached to \(c\), like subtraction or multiplication. Achieving isolated \(c\) simplifies the equation, revealing the solution.

Checking Solutions

After calculating a potential solution, we must verify our work. Checking solutions confirms whether our calculated value satisfies the original equation. Follow these steps:

• Substitute \(c = \frac{1}{6}\) back into the original expression: \(|6(\frac{1}{6}) - 1|\).
• Calculate the inside: \(|1 - 1|\ = |0|\ = 0\).

The calculated absolute value equals the right side of the equation. Therefore, our solution \(c = \frac{1}{6}\) is verified. This process reassures that we haven't made errors in earlier steps, and our solution is exactly what’s needed.