Problem 36 - Intermediate Algebra

Question

Solve each equation. See Example 2. $$ |8-5 x|-8=10 $$

Step-by-Step Solution

Answer

The solutions are \(x = -2\) and \(x = 5.2\).

Step 1: Isolate the absolute value

The equation is given as \(|8-5x| - 8 = 10\). To isolate the absolute value, first add 8 to both sides of the equation: \(|8-5x| = 18\).

Step 2: Consider the two cases

Since we have an absolute value equation, we need to consider both the positive and negative scenarios for the expression inside the absolute value: 1. \(8 - 5x = 18\) 2. \(8 - 5x = -18\)

Step 3: Solve the first case

Solve the equation \(8 - 5x = 18\) by subtracting 8 from both sides to get \(-5x = 10\). Then, divide both sides by -5, resulting in \(x = -2\).

Step 4: Solve the second case

Solve the equation \(8 - 5x = -18\) by subtracting 8 from both sides to obtain \(-5x = -26\). Then, divide both sides by -5 to get \(x = \frac{26}{5} = 5.2\).

Step 5: Check the solutions

Substitute \(x = -2\) back into the original equation to check: \(|8 - 5(-2)| - 8 = 10\) becomes \(|18| - 8 = 10\), which is true. Now substitute \(x = 5.2\): \(|8 - 5(5.2)| - 8 = 10\) becomes \(|8 - 26| - 8 = |18| - 8 = 10\), which is also true. Both solutions are correct.

Key Concepts

Absolute Value Equations

An absolute value equation is an equation that contains an absolute value expression. The absolute value of a number is its distance from zero on the number line, regardless of direction. It's always a non-negative value. For example, the absolute value \( |-3| \) is 3 because it's 3 units away from zero. Similarly, \( |3| \) is also 3.

When solving absolute value equations, it is important to remember that the expression inside the absolute value could be either positive or negative. This means we need to consider both scenarios when solving the equation. The solution will often involve two separate equations, one where the expression inside the absolute value equals a positive value, and the other where it equals the negative version of that value.

The general approach is:

• Isolate the absolute value expression on one side of the equation.
• Set the expression inside the absolute value equal to the positive and negative values of the other side.
• Solve both resulting equations to find all possible solutions.

Solving Equations

Solving equations is all about finding the value of the variable that makes the equation true. When it comes to equations with absolute values, you typically encounter two separate smaller tasks, corresponding to the scenarios inherent in absolute values.

For example, consider \( |8 - 5x| = 18 \). You need to resolve this into two distinct equations, \( 8 - 5x = 18 \) and \( 8 - 5x = -18 \). Why two? Because the expression inside the absolute value could be either 18 or -18 and still result in the absolute value being 18.

Once you have these equations, solve them in a straightforward manner:

• For \( 8 - 5x = 18 \), subtract 8 from both sides, obtaining \( -5x = 10 \). Then divide both sides by -5, resulting in \( x = -2 \).
• For \( 8 - 5x = -18 \), subtract 8 from both sides again to get \( -5x = -26 \). Divide both sides by -5, giving \( x = 5.2 \).

Step-by-Step Solutions

A step-by-step solution is an organized approach to solving a problem that breaks down the process into clear, manageable steps. This method is particularly helpfu for equations with more than one simple operation involved.

Here's how step-by-step solutions can demystify solving equations such as \( |8-5x| - 8 = 10 \):

• Step 1: Isolate the absolute value term. For the problem in question, you add 8 to both sides, resulting in \( |8-5x| = 18 \).
• Step 2: Recognize that there are two potential equations based on the absolute value function's nature. This gives you \( 8 - 5x = 18 \) and \( 8 - 5x = -18 \).
• Step 3: Solve each resulting equation independently, as shown in prior detailed steps, to find \( x = -2 \) and \( x = 5.2 \).
• Step 4: Verify each solution by plugging back into the original equation to ensure it satisfies the condition.

Using this clear, step-by-step method ensures that you don't miss any possible solutions and helps eliminate mistakes that often occur in more complex equations.