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Chapter 2: Problem 436

Solve. (a) \(|x|=7\) (b) \(|y|=-11\) (c) \(|z|=0\)

Short Answer

Expert verified
(a) x = 7 or x = -7, (b) No solution, (c) z = 0

Step by step solution

01

Understanding Absolute Value

Absolute value of a number represents its distance from zero on the number line, regardless of direction. This distance is always non-negative.
02

- Solve \(|x|=7\)

An absolute value equation \(|x|=a\) can be split into two separate equations: \(x = a\) and \(x = -a\). For \(|x|=7\), the solutions are: \(x = 7\) and \(x = -7\).
03

- Solve \(|y|=-11\)

Absolute value \(|y|\) cannot be negative, because distance cannot be negative. Therefore, there is no solution for the equation \(|y|=-11\).
04

- Solve \(|z|=0\)

Absolute value of zero is zero. Therefore, the only solution for \(|z|=0\) is \z = 0\.

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Key Concepts

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

solving absolute value equations
Absolute value equations often involve finding the unknown variable inside the absolute value symbol. The absolute value of a number is the distance between that number and zero on the number line, so it is always positive.

When solving \(|x| = a\) where \(a\) is a positive number, we split the equation into two possible scenarios:

  • The positive case: \(x = a\).
  • The negative case: \(x = -a\).

For example, solving \(|x| = 7\) involves setting up the equations \(x = 7\) and \(x = -7\), giving the solutions \(x = 7\) and \(x = -7\). However, if \(a\) is negative, such as in \(|y| = -11\), there are no solutions because absolute values cannot be negative.
non-negative distance
The concept behind absolute value is the idea of distance without direction. Mathematically, it's written as: \( |x| \). This symbol represents the non-negative distance of a number \(x\) from zero.

For any number \(x\):
  • If \(x\) is positive or zero, then \(|x| = x\).
  • If \(x\) is negative, then \(|x| = -x\), which turns \(x\)'s sign to positive.

For instance, the solutions to \(|z| = 0\) focus on finding the distance from zero. Since the only number with a distance of zero from zero is zero itself, the solution is simply \(z = 0\). This illustrates how the absolute value function centers around transforming any number into a non-negative distance.
absolute value properties
Absolute values have unique properties that help in simplifying and solving equations:

  • Non-Negativity: \(|x| \geq 0\) for any real number \(x\). This means the result of an absolute value function is always non-negative.
  • Identity: \(|x| = 0\) if and only if \(x = 0\). This shows that zero is the only number with an absolute value of zero.
  • Symmetry: \(|-x| = |x|\). This property indicates that the absolute value function treats positive and negative inputs symmetrically.
  • Triangle Inequality: \(|x + y| \leq |x| + |y|\). This property is useful in more advanced problem-solving and indicates that the absolute value of a sum is less than or equal to the sum of absolute values.

Understanding these properties helps in tackling diverse problems involving absolute values, making the solutions systematic and logical.

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