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Chapter 2: Problem 14

Solve each absolute value equation. See Examples 1 through 7. $$ |7 z|=0 $$

Short Answer

Expert verified
The solution is \(z = 0\).

Step by step solution

01

Understanding Absolute Value

The absolute value of a number refers to its distance from zero on a number line, regardless of direction. For any real number \(x\), \(|x|\) is non-negative, so \(|x| \geq 0\). Therefore, if \(|7z| = 0\), it implies \(7z\) must be zero.
02

Setting Up the Equation

Given the equation \(|7z| = 0\), we understand that the expression inside the absolute value (\(7z\)) must be equal to zero itself because the only number with an absolute value of zero is zero itself.
03

Solving for the Variable

To solve for \(z\), set \(7z = 0\). Solve this equation by dividing both sides by 7 to isolate \(z\). This results in \(z = 0\).

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Key Concepts

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

Equations
An equation is a mathematical statement that asserts the equality of two expressions. It consists of two sides, typically separated by an equal sign (=). In the context of absolute value equations, such as \(|7z| = 0\), the goal is to find the values of the variable that make the equation true.

Here are some key points to understand about equations:
  • An equation can have one or more solutions, or no solution at all, depending on its form.
  • Absolute value equations may require special attention because the absolute value operation affects potential solutions.
  • Learn to recognize different types of equations and apply appropriate methods for solving each type.
Understanding how to manipulate and balance equations is crucial for solving any mathematical problem.
Problem Solving
Problem solving is an essential skill in mathematics and everyday life. It involves finding solutions to complex or unfamiliar situations. With equations, like the one given in the problem \(|7z| = 0\), problem solving requires logical reasoning and analytical thinking.

Here's how to effectively tackle such problems:
  • Clearly understand the problem and identify what is being asked. In this case, determine what makes the absolute value zero.
  • Break down the problem into smaller, more manageable steps. For instance, establishing \(7z = 0\) after understanding absolute value properties.
  • Look for patterns or use prior knowledge to guide your solution.
  • Check the solution to ensure its validity within the context of the problem.
By practicing a structured approach to problem solving, students can improve their confidence and efficiency in math.
Real Numbers
Real numbers encompass all the numbers we typically encounter in mathematics, including both rational and irrational numbers. They can be visualized on the number line, spanning from negative infinity to positive infinity.

When dealing with absolute value equations, it is important to understand real numbers, as it helps to determine possible solutions:
  • Real numbers include integers, fractions, and decimals. For example, \(-3,\ \frac{1}{2},\ 4.5\) are all real numbers.
  • The absolute value of any real number is always non-negative. This means \(|x|\) is always \(\geq 0\).
  • Real numbers make it possible to place an absolute value equation like \(|7z|=0\) on the number line, where zero is the only point.
Having a grasp of real numbers helps when analyzing and interpreting mathematical problems, especially those involving absolute values.

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Most popular questions from this chapter

Write an absolute value inequality representing all numbers \(x\) whose distance from 0 is less than 7 units.

Solve each inequality. Graph the solution set and write it in interval notation. $$ 8+|x|<1 $$

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). $$ \begin{array}{|c|c|c|} \hline \text { Planet } & {\text { Miles from the Sun }} & {\text { AUfrom the Sun }} \\ \hline \text { Mercury } & {36 \text { million }} & {0.388} \\ \hline \end{array} $$ Jupiter 483.3 million

Write \(-5 \leq x \leq 5\) as an equivalent inequality containing an absolute value.

Solve each equation or inequality for \(x\). $$ |x+11|=-1 $$
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