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Chapter 2: Problem 31

Solve each absolute value equation. See Examples 1 through 9. $$ |6 x|-1=11 $$

Short Answer

Expert verified
The solutions are \(x = 2\) and \(x = -2\).

Step by step solution

01

Isolate the Absolute Value

Start by isolating the absolute value expression on one side of the equation. Add 1 to both sides to achieve this: \ \(|6x| - 1 + 1 = 11 + 1\) \ This simplifies to: \ \(|6x| = 12\)
02

Set Up Two Equations

When you have an equation of the form \ \(|A| = B\), it means that \ \(A = B\)and \ \(A = -B\). Apply this to \ |6x|=12\:\(6x = 12\) and \ \(6x = -12\).
03

Solve the First Equation

Solve the equation \ 6x = 12:\Divide both sides by 6: \ x = \frac{12}{6}\\(x = 2\)
04

Solve the Second Equation

Solve the equation \ 6x = -12:\Divide both sides by 6:\x = \frac{-12}{6}\\(x = -2\)
05

Write the Solution Set

The solutions to the absolute value equation \ |6x|-1 = 11\are the values obtained in steps 3 and 4. Thus, the solution set is \(x = 2\) or \ x = -2\.

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Key Concepts

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

Solving Absolute Value Equations
When faced with absolute value equations, the first step is often to isolate the absolute value expression. This means getting the expression by itself on one side of the equation. Think of it as a two-part unfolding process. Once the absolute value term like \(|6x|\) is isolated, you can create two separate equations. This is possible because the absolute value of a number represents its distance from zero, meaning \(6x\) could be either a positive or negative value that equals 12.
  • Set up two equations using the principle that if \(|A| = B\), then \(A = B\) or \(A = -B\).
  • This method allows us to determine all possible values for the variable, covering both scenarios where \(6x\) could result in a positive 12 or a negative 12.

This unfolding into two equations is key to solving absolute value equations, as it handles both potential outputs of the absolute value bracket.
Solutions to Equations
Once you've set up the two separate equations from the absolute value equation, the next step is to solve each one individually. Here, you are determining the specific numeric values that satisfy the original equation. Solving is straightforward:
  • For \(6x = 12\), divide by 6 to find \(x = 2\).
  • For \(6x = -12\), similarly divide by 6 to get \(x = -2\).

Each of these calculations gives you a solution to the equation, which means both \(x = 2\) and \(x = -2\) are valid solutions. Together, these form the solution set to the absolute value equation \(|6x|-1 = 11\). Remember that these solutions should always be checked back in the original equation to ensure their accuracy.
Absolute Value Properties
Understanding the properties of absolute values is essential in solving equations involving them. Absolute value, denoted by the vertical bars \(| |\), measures how far a number is from zero on a number line, irrespective of direction.
  • Key property: \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\).
  • This indicates that any number inside absolute value bars can either be positively expressed or negatively paired to produce the same output.

These properties mean that whenever you see an equation like \(|6x|\), it can represent either a positive \(6x\) or a negative \(-6x\), leading to a set of mirrored scenarios we explore in problem-solving. By grasping these principles, you gain the ability to navigate and solve absolute value equations more confidently.

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Most popular questions from this chapter

Solve each inequality. Graph the solution set and write it in interval notation. $$ |0.6 x-3|>0.6 $$

Solve each inequality. Graph the solution set and write it in interval notation. $$ |-1+x|-6>2 $$

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Solar system distances are so great that units other than miles or kilometers are often used. For example, the astronomical unit \((A U)\) is the average distance between Earth and the sun, or \(92,900,000\) miles. Use this information to convert each planet's distance in miles from the sun to astronomical units. Round to three decimal places. The planet Mercury's AU from the sun has been completed for you. (Source: National Space Science Data Center). $$ \begin{array}{|c|c|c|} \hline \text { Planet } & {\text { Miles from the Sun }} & {\text { AUfrom the Sun }} \\ \hline \text { Mercury } & {36 \text { million }} & {0.388} \\ \hline \end{array} $$ Mars \(\quad 141.5\) million
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