Problem 24 - Elementary and Intermediate Algebra

Question

Solve each equation. \(|5 x-3|=22\)

Step-by-Step Solution

Answer

The solutions are \( x = 5 \) and \( x = -\frac{19}{5} \).

Step 1: Understanding Absolute Value

We start by understanding that the equation involves an absolute value, which means \[ |5x-3| = 22 \]translates to two different possible equations: \[ 5x - 3 = 22 \] and \[ 5x - 3 = -22 \].

Step 2: Equation 1: Solving for Positive

For the first scenario:\[ 5x - 3 = 22 \]add 3 to both sides to get:\[ 5x = 25 \]Then divide both sides by 5:\[ x = 5 \]

Step 3: Equation 2: Solving for Negative

For the second scenario:\[ 5x - 3 = -22 \]add 3 to both sides to get:\[ 5x = -19 \]Then divide both sides by 5:\[ x = -\frac{19}{5} \]

Step 4: Solution Verification

Verify both solutions by plugging them back into the original absolute value equation to ensure they satisfy the equation.1. For \( x = 5 \):\[ |5(5) - 3| = |22| = 22 \] which is correct.2. For \( x = -\frac{19}{5} \):Calculate \[ |5(-\frac{19}{5}) - 3| = |-22| = 22 \] which is also correct. Therefore, both solutions are valid.

Key Concepts

Equation Solving

In the realm of absolute value equations, solving the equation involves breaking down the problem into simpler components. The absolute value notation \(|...|\) implies that the expression inside can be equal to either a positive or a negative value once the absolute values are removed. This tells us that the equation \(|5x - 3| = 22\) splits into two separate linear equations: \(5x - 3 = 22\) and \(5x - 3 = -22\).
To solve these linear equations:

• In equation one, \(5x - 3 = 22\), add 3 to both sides to isolate \(5x\), which results in \(5x = 25\). Divide by 5 to find \(x = 5\).
• In equation two, \(5x - 3 = -22\), add 3 to both sides to get \(5x = -19\). Dividing by 5 gives \(x = -\frac{19}{5}\).

These steps demonstrate how to break the absolute value equation into more manageable linear equations.

Verification of Solutions

After finding potential solutions to the equation, it's crucial to verify them by substituting back into the original equation \(|5x - 3| = 22\). This step ensures that both solutions accurately satisfy the absolute value condition.
Verification involves:

• Checking \(x = 5\): Substitute into the equation: \(|5(5) - 3| = |22| = 22\). The solution checks out.
• Checking \(x = -\frac{19}{5}\): Substitute and compute: \(|5(-\frac{19}{5}) - 3| = |-22| = 22\). This also confirms correctness.

By verifying both solutions, we ensure that each possible value for \(x\) is indeed a correct answer that adheres to the absolute value constraints.

Positive and Negative Scenarios

Absolute value equations like \(|5x - 3| = 22\) inherently produce both positive and negative solutions. Examining these scenarios separately is key in capturing all valid solutions.
Here's how they are treated:

• The **Positive Scenario**: Treat the absolute value as a straightforward equation, \(5x - 3 = 22\), yielding the solution \(x = 5\).
• The **Negative Scenario**: Relate the expression to its negative counterpart, \(5x - 3 = -22\), ensuring all roots are included with the solution \(x = -\frac{19}{5}\).

Recognizing both the positive and negative possibilities when solving absolute value equations is crucial for finding all potential solutions.