Problem 36 - Elementary and Intermediate Algebra

Question

Solve each equation. \(|8-5 x|-8=10\)

Step-by-Step Solution

Answer

The solutions are \(x = -2\) and \(x = \frac{26}{5}\).

Step 1: Isolate the Absolute Value

First, add 8 to both sides of the equation to isolate the absolute value expression:\[ |8 - 5x| - 8 + 8 = 10 + 8 \]This simplifies to:\[ |8 - 5x| = 18 \]

Step 2: Set Up Two Scenarios

Since the absolute value of an expression can be either the expression itself or the negative of the expression, set up two separate equations to consider both possibilities:1. \( 8 - 5x = 18 \)2. \( 8 - 5x = -18 \)

Step 3: Solve for x (First Scenario)

Solve the equation \(8 - 5x = 18\):Subtract 8 from both sides:\[ 8 - 8 - 5x = 18 - 8 \]This simplifies to:\[ -5x = 10 \]Divide both sides by -5:\[ x = -2 \]

Step 4: Solve for x (Second Scenario)

Solve the equation \(8 - 5x = -18\):Subtract 8 from both sides:\[ 8 - 8 - 5x = -18 - 8 \]This simplifies to:\[ -5x = -26 \]Divide both sides by -5:\[ x = \frac{26}{5} \]

Step 5: Check Solutions

Substitute \(x = -2\) back into the original equation:\[ |8 - 5(-2)| - 8 = 10 \]Calculate:\[ |8 + 10| - 8 = |18| - 8 = 10 \]True. Now check \(x = \frac{26}{5}\):\[ \left|8 - 5 \left(\frac{26}{5}\right)\right| - 8 = 10 \]Calculate:\[ |8 - 26| - 8 = |-18| - 8 = 10 \]Both solutions satisfy the original equation.

Key Concepts

Equation Solving

Equation solving is at the heart of algebra and involves finding the unknown variable that satisfies the given equation. In the case of an equation with an absolute value, like \(|8-5x|-8=10\), we need to first isolate the absolute value portion. Equation solving requires a strategic approach, and you often must perform operations such as addition or subtraction to move terms around, ensuring the expression containing the absolute value stands alone on one side of the equation.

This can be done by:

• Adding 8 to both sides to remove it from the left side, leaving \(|8 - 5x| = 18\).

Once isolated, the absolute value equation is broken into two separate linear equations. This is because the absolute value, by definition, gives us two scenarios to consider. Overall, the key is to carefully perform inverse operations step by step until the variable is isolated and the solutions verified.

This systematic approach ensures clarity and accuracy when determining the values of \(x\) that satisfy the equation.

Algebraic Concepts

Algebraic concepts are crucial in solving absolute value equations. The first concept is the absolute value itself, which represents the distance of a number from zero on a number line. It is always non-negative. In our equation, \(|8-5x|-8=10\), the absolute value \(|8-5x|\) must first be isolated so it can be accurately evaluated.

Understanding absolute values:

• The absolute value is represented as \(|a|\) and can be understood as:
  • \(a\) if \(a\geq0\)
  • \(-a\) if \(a<0\)

After isolating the absolute value, principles of linear equations help. We recognize that \(|8-5x|\) can equal either the expression itself \((8-5x = 18)\) or its negative \((8-5x = -18)\). These are the two scenarios you solve.

Such algebraic concepts are built step-by-step, helping to identify the nature of solutions and ensuring all possibilities are explored thoroughly.

Step-by-Step Solutions

Step-by-step solutions are an effective way to tackle complex equations by breaking them down into manageable parts. Solving the given absolute value equation involves a clear sequence of steps to ensure each aspect of the solution is understandable.

The first step is to isolate the absolute value expression, ensuring we deal with it cleanly. Once isolated, set up two scenarios to handle the nature of absolute values:

• First scenario: \(8 - 5x = 18\)
• Second scenario: \(8 - 5x = -18\)

These scenarios are then worked through independently by simplifying each to find the value of \(x\).

After solving each equation:

• \(x = -2\) for the first scenario
• \(x = \frac{26}{5}\) for the second scenario

Check each solution by substituting back into the original equation. This confirmation ensures that you have the correct solution, validating your work and building confidence in the process.

Step-by-step solutions shine as they break a problem down, making complex problems approachable and ensuring reliable results through systematic calculation.