Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|3 x+2|=7$$

Step-by-Step Solution

Answer

Based on the step-by-step solution, the solutions to the equation \(|3x+2|=7\) are \(x = 5/3\) and \(x = -3\).

Step 1: Understand the Problem

Given the equation \(|3x+2|=7\), when an absolute value is set equal to a positive number, as is the case here, there will be two resulting equations: \(3x + 2 = 7\) and \(3x + 2 = -7\).

Step 2: Solve the First Resulting Equation

Solving the equation \(3x + 2 = 7\) for \(x\), we subtract 2 from both sides to get \(3x = 5\). Then, by dividing both sides by 3, we obtain the solution \(x = 5/3\).

Step 3: Solve the Second Resulting Equation

Solving the equation \(3x + 2 = -7\) for \(x\), we subtract 2 from both sides to get \(3x = -9\). Then, dividing both sides by 3, we get the solution \(x = -9/3\) or \(x = -3\).

Step 4: Check the Solutions

By inserting \(x = 5/3\) into the original equation, we get \(|3 * 5/3 + 2| = 7\), which simplifies to \(|7|=7\), a true statement. Checking \(x = -3\), we get \(|3 * -3 + 2| = 7\), which simplifies to \(|7|=7\), also a true statement. So, both solutions are valid.

Key Concepts

Algebraic Solutions

Algebraic solutions involve breaking down an equation into manageable parts and systematically solving each part to find the values of the variable that satisfy the equation. In the case of absolute value equations, such as \(|3x+2|=7\), the absolute value defines a distance from zero, which means it can be equal to either a positive or negative number. Here are key steps to solving absolute value equations algebraically:

• Set the absolute value expression equal to both the positive and negative values of the constant on the other side of the equation.
• Solve each resulting equation separately to find potential solutions.

Breaking the equation into two equations, \(3x + 2 = 7\) and \(3x + 2 = -7\), allows us to consider both possibilities of the absolute value. Solving these equations algebraically will reveal all potential solutions.

Checking Solutions

Once you have potential solutions, it's crucial to verify them by substituting back into the original equation. This process helps ensure no errors were made during solving. Here's how you can check solutions for absolute value equations:

• Substitute each solution back into the original equation.
• Simplify to verify whether the left-hand side equals the right-hand side.
• If both sides match, the solution is valid. Otherwise, reconsider any possible errors in calculation or logic.

For instance, substituting \(x = \frac{5}{3}\) into \(|3x+2|=7\) simplifies to \(|7|=7\), confirming validity. Similarly, \(x = -3\) also simplifies correctly, ensuring it is also a valid solution. Hence, checking solutions is an essential step to confirm their legitimacy.

Solving Equations Step by Step

Solving equations step by step ensures clarity and accuracy in finding the solution. Let's break down the process for an equation involving absolute values:

• Understand the Equation: Recognize the structure of the absolute value to split the equation into two possible scenarios.
• Isolate the Absolute Value: Ensure that the absolute value is alone on one side of the equation to easily manage the calculations.
• Solve Each Equation: Create two equations from the absolute value where one considers the positive scenario (\(3x + 2 = 7\)) and the other, the negative scenario (\(3x + 2 = -7\)).
  • Example: For \(3x + 2 = 7\), subtract 2 from both sides, then divide by 3 to obtain \(x = \frac{5}{3}\).
  • Example: For \(3x + 2 = -7\), similarly subtract 2 and divide by 3 to find \(x = -3\).
• Check the Solutions: As previously stated, verify each result in the original equation for correctness.

Following such structured steps makes solving absolute value equations more intuitive and ensures all solutions are considered and verified.