Problem 15 - Algebra and Trigonometry

Question

Solve the equation. $$ 8+5\left|\frac{1}{3} x-\frac{5}{6}\right|=33 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{35}{2}\) and \(x = -\frac{25}{2}\).

Step 1: Isolate the Absolute Value Expression

To solve the equation \( 8 + 5\left|\frac{1}{3} x - \frac{5}{6}\right| = 33 \), start by subtracting 8 from both sides. This will help isolate the absolute value expression. \[5\left|\frac{1}{3} x - \frac{5}{6}\right| = 25\]

Step 2: Solve for the Absolute Value Expression

Next, divide both sides by 5 to completely isolate the absolute value:\[\left|\frac{1}{3} x - \frac{5}{6}\right| = 5\]

Step 3: Set Up Two Separate Equations

The absolute value equation \( \left|\frac{1}{3} x - \frac{5}{6}\right| = 5 \) translates into two separate linear equations:1. \( \frac{1}{3} x - \frac{5}{6} = 5 \)2. \( \frac{1}{3} x - \frac{5}{6} = -5 \)

Step 4: Solve the First Equation

Start solving the first equation: \[\frac{1}{3} x - \frac{5}{6} = 5\]Add \(\frac{5}{6}\) to both sides to isolate the term with \(x\):\[\frac{1}{3} x = 5 + \frac{5}{6} = \frac{30}{6} + \frac{5}{6} = \frac{35}{6}\]Now, multiply both sides by 3 to solve for \(x\):\[x = \frac{35}{2}\]

Step 5: Solve the Second Equation

Now solve the second equation:\[\frac{1}{3} x - \frac{5}{6} = -5\]Add \(\frac{5}{6}\) to both sides:\[\frac{1}{3} x = -5 + \frac{5}{6} = -\frac{30}{6} + \frac{5}{6} = -\frac{25}{6}\]Multiply both sides by 3 to solve for \(x\):\[x = -\frac{25}{2}\]

Step 6: Review Solutions

The solutions for \(x\) are \(\frac{35}{2}\) and \(-\frac{25}{2}\). These values satisfy the original equation. Verify these solutions by substituting them back into the original equation and confirming that both sides are equal.

Key Concepts

Understanding Linear Equations

Linear equations are fundamental for solving many types of mathematical problems, including those involving absolute values.
These equations are called "linear" because they graph as straight lines and have the general form of either \(ax + b = c\) or \(y = mx + c\).
To handle linear equations, you perform operations that maintain the equation's balance while working towards isolating the variable. Here, the key is to simplify the equation step by step to find the value of the unknown.

• We often begin by removing constants and coefficients surrounding the variable, typically through addition or subtraction.
• Next, divide or multiply to clear any remaining coefficients affecting the variable itself.

With the given problem, the goal after setting up the scenario is to first express two simpler linear equations from the absolute value part, paving the way for solving them through isolation and balance of terms.

Mastering Isolation of Variables

The concept of isolating variables is crucial when solving equations. It allows you to find the value that makes the equation true for the unknown variable.
When dealing with equations involving absolute values, the first step is often isolating the absolute value expression itself to simplify the problem.

• Start by performing algebraic operations to simplify the equation and isolate the absolute value term from constants and coefficients.
• In the original equation, subtracting 8 helped to reduce complexity and isolate the absolute value term.
• Following that, dividing by 5 removed the coefficient directly multiplying the absolute value term.

By completing these steps, you create a simpler absolute value equation that can be broken down into two linear equations, setting the stage for full isolation and solution of the variable in each scenario.

Steps for Verification of Solutions

Verification of solutions is the final step and a critical part of ensuring your solutions are correct. Just finding answers isn't enough; they must satisfy the original equation.
Verification involves plugging the computed solutions back into the original equation to confirm they hold true.
Here's how you can do it effectively:

• Take each solution you found and substitute it into the original equation.
• For the equation \(8 + 5\left|\frac{1}{3}x - \frac{5}{6}\right| = 33\), substitute \(x = \frac{35}{2}\) and \(x = -\frac{25}{2}\) back in.
• Compute each side of the equation to see if they are equal. If both sides match, the solution is verified.

This ensures that no errors occurred during the solving process and confirms the integrity of the solutions obtained.