Problem 59 - College Algebra with Corequisite Support

Question

Solve \(|3 x+1|=|2 x+3|\)

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -\frac{4}{5}\).

Step 1: Understand Absolute Value Equations

To solve the equation \(|3x+1|=|2x+3|\), we understand that this involves absolute values. The expression inside each absolute value can be either positive or negative when equal to one another. This means two potential cases for which we must solve.

Step 2: Set Up Case 1

For Case 1, assume both expressions inside the absolute value signs are equivalent as they are, yielding the equation: \[3x + 1 = 2x + 3\].

Step 3: Solve Case 1

Solving the equation from Case 1:1. Subtract \(2x\) from both sides:\[3x - 2x + 1 = 3\]2. Simplify to get:\[x + 1 = 3\]3. Subtract 1 from both sides:\[x = 2\].

Step 4: Set Up Case 2

For Case 2, assume the expressions inside the absolute values are opposites, leading to the equation: \[3x + 1 = -(2x + 3)\].

Step 5: Solve Case 2

Solving the equation from Case 2:1. Distribute the negative sign on the right side:\[3x + 1 = -2x - 3\]2. Add \(2x\) to both sides:\[3x + 2x + 1 = -3\]3. Combine like terms:\[5x + 1 = -3\]4. Subtract 1 from both sides:\[5x = -4\]5. Divide both sides by 5:\[x = -\frac{4}{5}\].

Step 6: Verify Solutions

Check both solutions \(x = 2\) and \(x = -\frac{4}{5}\) in the original equation to ensure they satisfy \(|3x+1|=|2x+3|\):1. For \(x = 2\): - \(|3(2)+1| = |7| = 7\) and \(|2(2)+3| = |7| = 7\), which holds true.2. For \(x = -\frac{4}{5}\): - \(|3(-\frac{4}{5})+1| = |-\frac{12}{5} + \frac{5}{5}| = |-\frac{7}{5}| = \frac{7}{5}\) - \(|2(-\frac{4}{5})+3| = |-\frac{8}{5} + \frac{15}{5}| = |\frac{7}{5}| = \frac{7}{5}\), which also holds true.Thus, both solutions are correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that allows us to find the unknown values in a given expression. When solving equations, our main goal is to isolate the variable, which will help us determine its value more easily. For absolute value equations, like \[ |3x + 1| = |2x + 3| \]we need to consider the properties of absolute values. Remember that absolute values make a number positive or leave it unchanged if it's already positive, which can create multiple scenarios or cases.
To solve an absolute value equation, both expressions inside the absolute values must be equal or opposites. Thus, it's essential to perform case analysis, setting up two different cases. Solve each case by simplifying both sides and ensuring that the variable is properly resolved to get meaningful solutions.
Another key aspect is that, after calculating possible solutions, we should always return to the original equation to verify their correctness. This prevents potential oversights, ensuring that all derived solutions are valid within the context of the initial mathematical statement.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations, such as addition, subtraction, multiplication, and division. They form the building blocks of equations in algebra. When dealing with expressions like \[ 3x + 1 \] and \[ 2x + 3 \]it's crucial to understand how to simplify and manipulate them to solve equations effectively.
One common method is to perform operations such as adding, subtracting, multiplying, or dividing both sides of an equation by the same quantity. This helps maintain equality while simplifying the expression.

• Combine like terms, such as merging all instances of the variable on one side.
• Use addition or subtraction to clear constant terms.
• Apply multiplication or division to isolate the variable.

In our example, simplifying the expression allows us to isolate \( x \) and find the real values that make the equation valid.
Each step requires careful arithmetic to ensure accuracy, helping us avoid errors in our solutions.

Case Analysis

Case analysis is a powerful method used to tackle absolute value equations by considering different scenarios within the equation. This technique involves splitting the equation into multiple cases to account for the multiple possibilities that absolute values present. Each case examines a unique situation, either where the expressions are equal or when they are opposites.For the equation \[ |3x+1| = |2x+3| \]we analyze two possible cases:

• Case 1: Both expressions inside the absolute values are equal: \[ 3x + 1 = 2x + 3 \]
• Case 2: Both expressions inside the absolute values are opposites: \[ 3x + 1 = -(2x + 3) \]

By solving each case separately, we derive potential solutions more systematically.
At the end of the process, it's crucial to check all solutions within the original equation, ensuring correctness and consistency in our results. This methodology ensures that no potential solutions are overlooked and that all cases are thoroughly considered.