Problem 77 - Algebra for College Students

Question

Solve each equation or inequality. Graph the solution set. $$ |2-0.2 x|=2 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 20\).

Step 1: Set Up the Equation

Given the equation \(|2-0.2 x|=2\). Since it is an absolute value equation, it can be split into two separate linear equations.

Step 2: Remove the Absolute Value

Remove the absolute value by considering both the positive and negative cases. Set up two equations: \((2 - 0.2x) = 2\) and \((2 - 0.2x) = -2\).

Step 3: Solve the First Equation

Solve \((2 - 0.2x) = 2\). Subtract 2 from both sides to get \(-0.2x = 0\). Divide by \(-0.2\) to find \(x = 0\).

Step 4: Solve the Second Equation

Solve \((2 - 0.2x) = -2\). Subtract 2 from both sides to get \(-0.2x = -4\). Divide by \(-0.2\) to find \(x = 20\).

Step 5: Graph the Solution Set

Plot the solution set on a number line. The solutions are \(x = 0\) and \(x = 20\). Draw points at \(x = 0\) and \(x = 20\).

Key Concepts

Solving Absolute Value Equations

Absolute value equations can look tricky at first. However, they become simpler when you know how to approach them. Absolute value measures the distance between a number and zero on a number line. This distance is always positive or zero. For the equation \(|2 - 0.2x| = 2\), we start by understanding that the expression inside the absolute value can be either positive or negative. This means we need to consider two scenarios: \(2 - 0.2x = 2\) and \(2 - 0.2x = -2\).Steps to Remove the Absolute Value:

• Write two separate equations: one for the positive value and one for the negative value.
• Solve each equation independently.

Step-by-step, this method ensures you account for all possible values of \(x\) that make the equation true.

Linear Equations

While solving absolute value equations, you'll work with linear equations. Linear equations are equations where the highest power of the variable (usually \(x\)) is one. They have a simple form: \(ax + b = c\). In our absolute value problem, after splitting it, we got two linear equations to solve:

• \(2 - 0.2x = 2\)
• \(2 - 0.2x = -2\)

Solving These Linear Equations:

• For \(2 - 0.2x = 2\), subtract 2 from both sides to get \(-0.2x = 0\). Then divide by \(-0.2\) to find \{x = 0\}.
• For \(2 - 0.2x = -2\), subtract 2 from both sides to get \(-0.2x = -4\). Then divide by \(-0.2\) to find \{x = 20\}.

After solving, you find that \(x = 0\) and \(x = 20\) meet the original equation.

Graphing Solutions on a Number Line

Once you've solved the equations, it's essential to visualize the solutions. A helpful method is graphing on a number line. This gives a clear picture of where the solutions lie.Steps to Graph Solutions:

• Draw a horizontal line (the number line).
• Mark the positions of your solutions, \(x = 0\) and \(x = 20\), on the line.
• Place a dot or circle at both points to indicate solutions.

Graphing can make abstract solutions much more concrete, making it easier to understand where solutions fit in comparison to each other. Here, both points (0 and 20) show the values for \(x\) that satisfy our original absolute value equation.