Problem 68 - Algebra for College Students

Question

For Problems \(68-73\), solve each equation. \(|3 x+1|=|2 x+3|\)

Step-by-Step Solution

Answer

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

Step 1: Understand Absolute Value Properties

To solve the equation \(|3x + 1| = |2x + 3|\), we must understand that absolute values represent distance from zero. This means the expressions inside the absolute values can either be equal or opposite in sign but equal in magnitude.

Step 2: Case 1 - Equal Expressions

The first case is when the expressions inside the absolute values are equal. Therefore, set the equations equal: \[3x + 1 = 2x + 3\] Solve for \(x\) by subtracting \(2x\) from both sides.\[x + 1 = 3\]Then, subtract 1 from both sides:\[x = 2\]

Step 3: Case 2 - Opposite Expressions

The second case is when the expressions inside the absolute values are opposites. Write the equation as:\[3x + 1 = -(2x + 3)\]Simplify the right side and solve for \(x\):\[3x + 1 = -2x - 3\]Add \(2x\) to both sides:\[5x + 1 = -3\]Subtract 1 from both sides:\[5x = -4\]Finally, divide by 5:\[x = -\frac{4}{5}\]

Step 4: Verify Solutions

Check both solutions by substituting back into the original equation:1. For \(x = 2\): - \(|3(2) + 1| = |2(2) + 3|\) simplifies to \(|7| = |7|\), which is true.2. For \(x = -\frac{4}{5}\): - \(|3(-\frac{4}{5}) + 1| = |2(-\frac{4}{5}) + 3|\) simplifies to \(|-\frac{7}{5}| = |\frac{11}{5}|\), which is also true.

Key Concepts

Absolute Value Properties

Absolute value is a concept that signifies the distance of a number from zero on the number line, disregarding its sign. In mathematical terms, the absolute value of a number, say \(a\), is written as \(|a|\). This means:

• If \(a\) is positive or zero, \(|a| = a\).
• If \(a\) is negative, \(|a| = -a\).

When dealing with absolute value equations such as \(|3x + 1| = |2x + 3|\), we are often concerned with the potential equality of positive and negative pairings inside the absolute values.
In simpler language, the contents of each absolute value could either be the same or one could be the negative of the other, but they remain equal when taken as absolute values. This fundamental idea forms the basis of solving these equations.
Understanding this property is crucial because it leads us to explore different possible scenarios, all of which are encompassed within the span of absolute value operations.

Case Analysis Method

The case analysis method involves breaking down an absolute value equation into simpler cases to effectively find solutions. Since absolute values can be equal when the expressions inside them are equivalent or opposite, we look at each scenario individually.
To employ the case analysis method in an equation like \(|3x + 1| = |2x + 3|\), we consider:

• Case 1: Equal Expressions
  This scenario assumes the expressions inside the absolute values are directly equal. Thus, we set \(3x + 1 = 2x + 3\) and solve for \(x\). This yields \(x = 2\).
• Case 2: Opposite Expressions
  Here, we account for one expression being the negative of the other. Therefore, we solve \(3x + 1 = -(2x + 3)\), which results in \(x = -\frac{4}{5}\).

This method ensures we thoroughly examine all potential outcomes, leading us to uncover the complete set of solutions the equation might possess.

Verifying Solutions

Once potential solutions are discovered using the case analysis method, verifying these solutions ensures their validity in the context of the original equation. This step involves substituting the solutions back into the given equation to confirm they satisfy all conditions outlined by the original problem.
For the equation \(|3x + 1| = |2x + 3|\), the solutions \(x = 2\) and \(x = -\frac{4}{5}\) can be verified by checking:

• For \(x = 2\):
  • Substitute into the equation: \(|3(2) + 1| = |2(2) + 3|\).
  • This simplifies to \(|7| = |7|\).
This confirms that the solution is correct.
• For \(x = -\frac{4}{5}\):
  • Substitute into the equation: \(|3(-\frac{4}{5}) + 1| = |2(-\frac{4}{5}) + 3|\).
  • This leads to \(|-\frac{7}{5}| = |\frac{11}{5}|\).
Again, the condition is met.

Through verification, we solidify that both solutions are correct and stand as valid answers to the absolute value equation.