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Chapter 2: Problem 43

For the following exercises, solve for the unknown variable. $$ \sqrt{|x|^{2}}=x $$

Short Answer

Expert verified
The solution is \(x \geq 0\).

Step by step solution

01

Recognize the Absolute Value Property

The expression under the square root, \(|x|^2\), simplifies to \(|x|\) because the square of any real number is non-negative and its square root is the absolute value. Thus, the equation becomes \(|x| = x\).
02

Consider the Definition of Absolute Value

The absolute value \(|x|\) means x can either be positive or zero. Hence, \(|x| = x\) implies that \(x\) must be non-negative because only when \(x \geq 0\), \(|x|\) returns the original value of \(x\).
03

Solve for x

Since \(|x| = x\) implies \(x \geq 0\), we can conclude \(x\) must be a non-negative number. To check this condition, it's clear that both sides of the equation are equal only if \(x \leq 0\) results in contradiction (since square roots are non-negative). Hence, \(x\) must be non-negative.

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Key Concepts

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

Square Root Property
The Square Root Property is a useful tool in solving equations that involve squared terms. It states that if a squared term equals a number, then the original term is either the positive or negative square root of that number. However, when dealing with absolute value equations like \( \sqrt{|x|^2} = x \), this property helps simplify the equation by eliminating the power and square root. In such cases,
  • the squared term \(|x|^2\) simplifies to \(|x|\), because the square of absolute value will be non-negative, and taking the square root of a non-negative number results in its absolute value.
  • this simplification into \(|x| = x\) avoids dealing with both positive and negative roots explicitly because absolute values are inherently non-negative.
Understanding this property helps us focus on the non-negative nature of the absolute value in simplifying the equation further.
Solving Equations
Solving equations, especially those involving absolute values and square roots, requires careful handling of the properties involved. In the equation \(\sqrt{|x|^2} = x \), the primary approach is simplification using the Square Root Property. Let's break it down:
  • First, recognize the expression \(|x|^2\) under the square root, which simplifies to \(|x|\).

  • The equation is transformed from having a square root to a straightforward absolute value equation, \(|x| = x\).

  • Use the definition of absolute value, which states \(|x|\) equals \(x\) if \(x\) is positive or zero.
By focusing on each step and applying these strategies, solving such equations becomes a more manageable process.
Real Numbers
Real numbers form the basis of comprehending equations like \(\sqrt{|x|^2} = x\). These numbers include all possible values that are either positive, negative, or zero, but here, only the non-negative are significant.
  • When considering the simplification \(|x|^2 = |x|\), realize real numbers always yield non-negative squares, crucial to solving this type of equation.

  • The equation essentially limits \(x\) to real numbers that satisfy \(x \geq 0\) since only non-negative values fit the absolute value condition \(|x| = x\).
Therefore, understanding real numbers and their properties allow us to identify solutions that suit the absolute value's inherent nature.
Non-negative Values
Non-negative values are central to the equation \(|x| = x\) derived from \(\sqrt{|x|^2} = x\). Non-negative numbers include zero and all positive numbers. Here's how they fit into solving the equation:
  • Absolute values, by definition, are always non-negative, which implies \(x\) must be non-negative to satisfy \(|x| = x\).

  • When \(x\) is positive or zero, both sides of the equation align perfectly because \(|x|\) holds no other value than \(x\) itself.

  • If \(x\) were negative, it would contradict the absolute value definition because \(|x|\) could not equal a negative \(x\).
This condition ensures we maintain logical consistency, confirming that solutions are only possible for non-negative \(x\).

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Most popular questions from this chapter

For the following exercises, input the left-hand side of the inequality as a \(Y 1\) graph in your graphing utility. Enter \(y 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, \(1: a b s(.\) Find the points of intersection, recall \(\left(2^{\text {nd }}\right.\) CALC 5 :intersection, \(1^{\text {st }}\) curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x+2|-5<2 $$

For the following exercises, write the set in interval notation. $$ \\{x \mid-1

For the following exercises, solve the inequality. Write your final answer in interval notation. $$ -2 x+3>x-5 $$

For the following exercises, solve the equation involving absolute value. $$ |2 x+1|-2=-3 $$

For the following exercises, write the interval in set-builder notation. [-4,1]\(\cup[9, \infty)\)
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