Problem 40 - Algebra 1: Concepts and Skills

Question

Use mental math to solve the equation. If there is no solution, write no solution. $$ |x|=\frac{5}{6} $$

Step-by-Step Solution

Answer

The equation |x| = \(\frac{5}{6}\) has two solutions: x = \(\frac{5}{6}\) and x = -\(\frac{5}{6}\)

Step 1: Understanding absolute value

An absolute value denotes the distance of a number from zero, without considering direction. Therefore, \(|x|\) could either be a positive value or zero. If \(|x| = \frac{5}{6}\), it means x could either be \(\frac{5}{6}\) or \(-\frac{5}{6}\)

Step 2: Solving for positive value

Let's first solve for the positive value where x equals to \(\frac{5}{6}\). Simply equating the expressions, we can say, x = \(\frac{5}{6}\)

Step 3: Solving for negative value

Now, let's solve for the negative value where x could be \(-\frac{5}{6}\). Equating the expressions, we get, x = -\(\frac{5}{6}\)

Key Concepts

Understanding Mental Math for Absolute Value Equations

Solving equations using mental math involves performing calculations in your head. This approach can be particularly useful when dealing with absolute values since it can simplify the process and save time.
When we look at the equation \(|x| = \frac{5}{6}\), our mental math skills tell us that we're searching for numbers whose distance from zero is \(\frac{5}{6}\).
Since the absolute value reflects this distance without any regard for direction, it suggests two scenarios:

• The number is \(\frac{5}{6}\).
• The number is \(-\frac{5}{6}\).

By mentally processing these possibilities, we can confidently determine the values of \(x\) without having to resort to elaborate calculations or written work. This practice enhances our intuition about numbers and strengthens our problem-solving skills.

Deep Dive into Solving Equations with Absolute Values

When solving absolute value equations, such as \(|x| = \frac{5}{6}\), it's important to understand what the absolute value represents. The key here is to appreciate that the absolute value measures a number's distance from zero on the number line.
This means that both positive (\(x = \frac{5}{6}\)) and negative values (\(x = -\frac{5}{6}\)) satisfy the equation because both are at an equal distance from zero, exemplified by this equation.
To solve such equations:

• Identify the given absolute value expression.
• Set up two separate equations to reflect both possible values, positive and negative.
• Solve each equation for \(x\).

This approach ensures that all potential solutions are considered, giving a full solution to the problem without overlooking any possibilities.

Exploring Positive and Negative Solutions

The equation \(|x| = \frac{5}{6}\) invites us to explore the concepts of both positive and negative solutions.
The absolute value equation tells us that \(x\) could be the positive number \(\frac{5}{6}\) because \(\frac{5}{6}\) units away from zero on the number line is \(\frac{5}{6}\) itself.
Simultaneously, \(x\) could also be the negative number \(-\frac{5}{6}\) because \(-\frac{5}{6}\) is also \(\frac{5}{6}\) units away from zero.Here's a helpful concept to remember:

• Absolute value represents a number's distance from zero, ignoring whether the number is negative or positive.
• For any equation of the form \(|x| = a\), the solutions are \(x = a \) and \(x = -a\).
• This means always considering both the positive and negative scenarios.

Understanding this ensures that no matter the distance specified by the absolute value, you consider both directions around zero, ensuring complete analysis.