Problem 75 - Algebra for College Students

Question

Solve each equation or inequality. Graph the solution set. $$ 3|x-6|=9 $$

Step-by-Step Solution

Answer

The solutions are x=3 and x=9.

Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression on one side of the equation. For the equation \[3|x-6|=9\], divide both sides by 3 to get \(|x-6|=3\).

Step 2: Set Up Two Separate Equations

Since \(|x-6|=3\), this means that the expression inside the absolute value can be equal to 3 or -3. Therefore, we set up the two equations: \(x-6=3\) and \(x-6=-3\).

Step 3: Solve Each Equation

Solve each of the two equations separately:1. For \(x-6=3\), add 6 to both sides to get \(x=9\).2. For \(x-6=-3\), add 6 to both sides to get \(x=3\).

Step 4: Provide the Solution Set

Combine the solutions from both equations to get the solution set. The solutions are \(x=9\) and \(x=3\). So, the solution set is \(\{3, 9\}\).

Step 5: Graph the Solution Set

Graph the solution set on a number line. Mark points at x=3 and x=9 and indicate these are solutions by highlighting or drawing circles around these points.

Key Concepts

absolute value equations

Absolute value equations involve expressions within absolute value bars, like \(|x-6|\).
The absolute value of a number is its distance from 0 on a number line. This distance is always positive or zero.
For example, \(|-3| = 3\) and \(|3| = 3\).
When solving absolute value equations, the goal is to isolate the absolute value expression and then solve the resulting equations.
The important steps are:

• Isolate the absolute value expression
• Set up two separate equations, one for the positive case and one for the negative case.
• Solve each equation.
• Combine solutions to form the solution set.

solution set

The solution set is the collection of all possible solutions to an equation or inequality.
For an absolute value equation \(|x-6| = 3\), the solution set includes all values of \(x\) that satisfy the equation.
In our example, starting with the isolated absolute value expression \(|x-6|=3\), we derive two separate equations:

• \(x-6 = 3\), which simplifies to \(x = 9\).
• \(x-6=-3\), which simplifies to \(x = 3\).

Hence, the solution set of the equation is \(\{3, 9\}\).
This means both \(x = 3\) and \(x = 9\) make the original equation true.

graphing solutions

Graphing solutions helps visualize where the solutions lie on the number line.
It provides a clear method to identify solution sets easily.
For the solution set \(\{3, 9\}\), we graph the solutions as follows:

• Draw a number line.
• Locate the points \(x = 3\) and \(x = 9\) on the number line.
• Mark these points with circles or dots to indicate they are solutions.

This visual representation confirms that both 3 and 9 are solutions to the absolute value equation \(|x-6|=3\).