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Chapter 1: Problem 75

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

Short Answer

Expert verified
The solutions are x=3 and x=9.

Step by step solution

01

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 \(|x-6|=3\).
02

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 \(x-6=-3\).
03

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 \(x=3\).
04

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 \(\{3, 9\}\).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(|x-6|=3\).

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