Problem 37 - Elementary and Intermediate Algebra

Question

Solve each equation. \(\left|\frac{7}{8} x+5\right|-2=7\)

Step-by-Step Solution

Answer

The solutions are \( x = \frac{32}{7} \) and \( x = -16 \).

Step 1: Isolate the Absolute Value

To start solving the equation \( \left|\frac{7}{8} x + 5\right| - 2 = 7 \), first isolate the absolute value expression by adding 2 to both sides of the equation:\[ \left| \frac{7}{8} x + 5 \right| = 9 \]

Step 2: Remove the Absolute Value

The equation \( \left| \frac{7}{8} x + 5 \right| = 9 \) means that the expression inside the absolute value can be either 9 or -9. So, set up two separate equations:1. \( \frac{7}{8} x + 5 = 9 \)2. \( \frac{7}{8} x + 5 = -9 \)

Step 3: Solve the First Equation

Start with the first equation: \( \frac{7}{8} x + 5 = 9 \).Subtract 5 from both sides to get:\[ \frac{7}{8} x = 4 \]Next, solve for \( x \) by multiplying both sides by \( \frac{8}{7} \):\[ x = \frac{8}{7} \cdot 4 = \frac{32}{7} \]

Step 4: Solve the Second Equation

Now solve the second equation: \( \frac{7}{8} x + 5 = -9 \).Subtract 5 from both sides to get:\[ \frac{7}{8} x = -14 \]Then solve for \( x \) by multiplying both sides by \( \frac{8}{7} \):\[ x = \frac{8}{7} \cdot (-14) = -16 \]

Step 5: State the Solutions

The solutions to the equation \( \left| \frac{7}{8} x + 5 \right| - 2 = 7 \) are the two values we found:\( x = \frac{32}{7} \) and \( x = -16 \).

Key Concepts

Absolute Value Equations

Absolute value equations represent a unique challenge because they include the concept of "distance" from zero. The absolute value of a number is its non-negative value, which makes solving these equations a bit different compared to standard ones. In an equation like \( \left|\frac{7}{8} x + 5\right| - 2 = 7 \), the expression inside the absolute value \( \left|\frac{7}{8} x + 5\right| \) needs to be isolated first, as shown in the step-by-step solution.The key to solving absolute value equations is understanding that the expression inside the absolute value, \( \frac{7}{8} x + 5 \), can be equal to a positive number and its negative counterpart. This is because the absolute value expression equates equal distance on both sides of zero, hence leading us to form two separate linear equations \( \frac{7}{8} x + 5 = 9 \) and \( \frac{7}{8} x + 5 = -9 \) after isolating the absolute value part. Solving these linear equations gives us the set of possible solutions.

Step-by-Step Problem Solving

Approaching algebra problems with a step-by-step method helps in organizing thoughts and systematically finding a solution. This approach is particularly useful for equations involving absolute values since they often lead to solving multiple equations. Here is a structured way to approach such problems:

• First, isolate the absolute value expression. This helps in simplifying the problem and sets a clear foundation for forming the subsequent equations.
• Translate the absolute value equation into two separate linear equations. This involves breaking down the lived condition, where the content inside could be positive or negative.
• Solve each equation individually. This part involves standard methods for solving linear equations, like simplifying expressions and isolating the variable.
• Finally, combine results and check if any solutions need to be dismissed. Having a checklist helps ensure you don't overlook potential results due to simplification errors.

By consistently using this step-by-step methodology, students can build confidence and accuracy in solving algebraic equations.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. A linear equation typically involves an unknown, often denoted by \( x \), that you solve for by isolating it on one side using algebraic operations.Let's break down the linear equations from our original problem:1. For the equation \( \frac{7}{8} x + 5 = 9 \): - Subtract 5 from both sides to eliminate the constant term. - Multiply by the reciprocal of \( \frac{7}{8} \), which is \( \frac{8}{7} \), to solve for \( x \). - This approach helps to undo the operations sequentially until \( x \) is isolated.2. Similarly, for \( \frac{7}{8} x + 5 = -9 \): - Again, begin by subtracting 5 from both sides. - Follow by multiplying by \( \frac{8}{7} \).Using these steps makes the solving direct and maintainable. It is helpful to write down each step to track your work and avoid mistakes. Familiarity with these steps for solving linear equations is essential, as they are foundational for numerous other algebraic processes.