Problem 17 - Algebra and Trigonometry

Question

Solve the equation. $$ |x-1|=|3 x+2| $$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{3}{2}\) and \(x = -\frac{1}{4}\).

Step 1: Understanding Absolute Value Equation

The equation given is \(|x-1| = |3x+2|\). Absolute value expressions can be solved by considering both the positive and negative scenarios.

Step 2: Case 1: Positive Scenario

In the positive scenario, assume both expressions inside the absolute values are positive or equal. This means we solve \(x-1 = 3x+2\).

Step 3: Solve Equation for Case 1

Starting with \(x - 1 = 3x + 2\), subtract \(x\) from both sides to get \(-1 = 2x + 2\). Then, subtract \(2\) from both sides to obtain \(-3 = 2x\). Finally, divide both sides by \(2\) to find \(x = -\frac{3}{2}\).

Step 4: Case 2: Negative Scenario

In the negative scenario, assume one expression inside the absolute value is negative while the other is positive. Consider \(x-1 = -(3x+2)\).

Step 5: Solve Equation for Case 2

Starting with \(x - 1 = -3x - 2\), add \(3x\) to both sides to get \(4x - 1 = -2\). Then, add \(1\) to both sides for \(4x = -1\). Divide by \(4\) to solve for \(x = -\frac{1}{4}\).

Step 6: Verification of Solutions

Verify the solutions \(x = -\frac{3}{2}\) and \(x = -\frac{1}{4}\) into the original equation \(|x-1| = |3x+2|\). Both satisfy the equation: \(|-\frac{5}{2}| = |-\frac{7}{2}|\) and \(|-\frac{5}{4}| = |-\frac{1}{4}|\) respectively.

Key Concepts

Positive and Negative Scenarios

When solving absolute value equations, it is important to consider all possible cases. The expression \(|x-1| = |3x+2|\) implies that the contents within each absolute value could be either both positive, both negative, or one positive and the other negative. Understanding these scenarios aids in crafting the correct method to solve the equation.

• **Positive scenario:** Both expressions \((x-1)\) and \((3x+2)\) are either equal or positive.
• **Negative scenario:** One expression is positive while the other is negative, creating an opposite balance.

This method ensures that we do not miss any potential solution that cannot be seen initially. By addressing each case separately, we ensure that all possible values for \(x\) are methodically examined.
Understanding these scenarios is key because they define the domain over which different expressions can operate, covering all potential outcomes.

Solving Linear Equations

Once the scenarios are clearly laid out, the next step involves solving the linear equations formed in each case. Let's delve into the steps needed for solving these equations.

In the *positive scenario*, we form the equation \(x-1 = 3x+2\). To solve, manipulate the terms to isolate \(x\). This involves:

• Subtracting \(x\) from both sides to get \(-1 = 2x + 2\).
• Subtracting \(2\) from both sides to result in \(-3 = 2x\).
• Finally, dividing both sides by \(2\) to solve for \(x = -\frac{3}{2}\).

In the *negative scenario*, we use \(x-1 = -(3x+2)\). The steps are similar, yet different equations:

• Add \(3x\) to both sides yielding \(4x - 1 = -2\).
• Add \(1\) to both sides to have \(4x = -1\).
• Finally, divide by \(4\) to find \(x = -\frac{1}{4}\).

These algebraic manipulations are common when solving linear equations. Practicing these techniques will help streamline the problem-solving process.

Verification of Solutions

Verification is a crucial step after obtaining solutions in any equation, especially for absolute value problems. It ensures that the derived solutions actually satisfy the original equation.

For our solutions, \(x = -\frac{3}{2}\) and \(x = -\frac{1}{4}\), we plug them back into the equation \(|x-1| = |3x+2|\) to verify:

• For \(x = -\frac{3}{2}\): Substitute into the equation:
  \(|-\frac{5}{2}| = |\frac{-9 + 2}{2}| = |-\frac{7}{2}|\)
  Both sides equate to \(\frac{5}{2}\), thus verifying this is a correct solution.
• For \(x = -\frac{1}{4}\): Again, substitute:
  \(|-\frac{5}{4}| = |-\frac{1}{4}|\)
  Both calculate to \(\frac{5}{4}\), confirming \(x = -\frac{1}{4}\) as a valid solution.

This step is essential to confirm our solutions' correctness since errors might occur during calculation. Verifying ensures reliability and builds confidence in the problem-solving process.