Problem 78 - Algebra for College Students

Question

Solve each equation or inequality. Graph the solution set. $$ |5-0.5 x|=4 $$

Step-by-Step Solution

Answer

The solutions are x = 2 and x = 18.

Step 1: Understand the Absolute Value Equation

An absolute value equation |A| = B can be split into two separate equations: A = B and A = -B. Here, we need to solve for the variable x using these properties. So we have two equations: 5 - 0.5x = 4 and 5 - 0.5x = -4.

Step 2: Solve the First Equation

For the first equation: 5 - 0.5x = 4. Subtract 5 from both sides to get: -0.5x = -1. Then divide by -0.5: x = 2.

Step 3: Solve the Second Equation

For the second equation: 5 - 0.5x = -4. Subtract 5 from both sides to get: -0.5x = -9. Then divide by -0.5: x = 18.

Step 4: Graph the Solution Set

The solutions of the equation are x = 2 and x = 18. On a number line, place points at x = 2 and x = 18 to represent the solution set.

Key Concepts

Solving equations

Solving equations means finding the value(s) of the variable that make the equation true. In our problem, we are solving the absolute value equation \(|5 - 0.5x| = 4\). The first step is to understand that an absolute value equation \(|A| = B\) can be split into two cases: \(A = B\) and \(A = -B\). This method helps us handle the absolute value and solve for the variable inside it.

Graphing solutions

Once we solve the equations, we need to graph the solutions to visualize them. For our equations, the solutions are \(x = 2\) and \(x = 18\). To graph these solutions, we use a number line. A number line is a horizontal line where each point represents a real number. Place a dot at \(x = 2\) and another dot at \(x = 18\) on the line. These points represent the solutions to the absolute value equation.

Algebraic manipulation

Algebraic manipulation involves using algebraic methods to rearrange and solve equations. For the first equation \((5 - 0.5x = 4)\), we subtract 5 from both sides to get \(-0.5x = -1\). Then, we divide by -0.5 to solve for x, resulting in \( x = 2 \). For the second equation \((5 - 0.5x = -4)\), we again subtract 5 from both sides, resulting in \(-0.5x = -9\), and dividing by -0.5 gives us \( x = 18 \). Both solutions must be checked back into the original absolute value equation to ensure they’re correct.

Absolute value properties

Absolute value represents the distance of a number from zero on a number line, regardless of direction. Key properties of absolute values are:

• \(|A| \geq 0 \): Absolute value is always non-negative.
• \(| A | = | -A | \): Both a number and its negative have the same absolute value.
• An absolute value equation \( | A | = B \) can be written as \( A = B \) and \( A = - B \).

Knowing these properties helps us understand and solve equations involving absolute values efficiently.