Problem 8 Solve. $$|x|=12$$... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 8

Solve. $$|x|=12$$

Short Answer

Expert verified

The solutions are $x = 12$ and $x = -12$.

Step by step solution

- Understand the Absolute Value Equation

The absolute value of a number is the distance of the number from zero on the number line, regardless of direction. Therefore, the equation $|x| = 12$ means that the value of $|x|$ is 12.

- Set Up Two Equations

The equation $|x| = 12$ can be broken down into two separate equations because absolute value represents distance in both positive and negative directions. These equations are: $x = 12$ and $x = -12$.

- Solve Each Equation

Solve both equations individually: $x = 12$ and $x = -12$.

- State the Solutions

Combine the solutions from both equations. The solutions to the equation $|x| = 12$ are $x = 12$ and $x = -12$.

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Key Concepts

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

Distance on Number Line

To fully grasp absolute value equations, it's crucial to understand the concept of 'distance on number line'. Absolute value essentially measures how far a number is from zero on a number line, without considering direction. For example, the absolute value of both 12 and -12 is 12. That鈥檚 because both 12 and -12 are 12 units away from zero, despite being in opposite directions. This 'distance' idea helps to visualize why $|x| = 12$ translates to two different values: 12 and -12.

Positive and Negative Solutions

One of the key steps in solving absolute value equations is recognizing that there can be both positive and negative solutions. Given $|x| = 12$, we know the 'distance' or absolute value is 12. This means the distance could be 12 units to the right (positive), which gives us the equation $x = 12$. Or, it could be 12 units to the left (negative), giving us $x = -12$. This dual approach is essential in solving such equations effectively.

Absolute Value Properties

Absolute value has special properties that often make solving equations easier:
$\bullet$ The absolute value of any real number is always non-negative.
$\bullet$ For any real number $x$, $|x| \geq 0$.
$\bullet$ $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$.
These properties are why $|x| = 12$ leads to both \x = 12\ and \x = -12\. Understanding these basics will help you approach and break down absolute value problems with more confidence.

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91影视

Solve. $$|x|=12$$

Short Answer

Step by step solution

- Understand the Absolute Value Equation

- Set Up Two Equations

- Solve Each Equation

- State the Solutions

Key Concepts

Distance on Number Line

Positive and Negative Solutions

Absolute Value Properties

One App. One Place for Learning.

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