Problem 43 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution is all non-negative values of \(x\), i.e., \(x \geq 0\).

Step 1: Understanding the absolute value

The absolute value of a number is always non-negative, which means that \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\).

Step 2: Evaluating the expression \(\sqrt{|x|^{2}}\)

Notice that \(|x|^{2} = x^2\), so \(\sqrt{|x|^{2}} = \sqrt{x^{2}}\), which simplifies to \(|x|\) since the square root of a square returns the absolute value.

Step 3: Setting up the equation

The equation given in the problem is \(\sqrt{|x|^{2}} = x\). After the simplification, the equation becomes \(|x| = x\).

Step 4: Solving the equation \(|x| = x\)

The equation \(|x| = x\) implies that \(x \geq 0\). Therefore, any non-negative value for \(x\) will satisfy the equation.

Step 5: Verifying the solution

For \(x \geq 0\), \(|x| = x\) is true, which satisfies the original equation. However, if \(x < 0\), \(|x| eq x\) because \(|x| = -x\), contradicting the equation. Therefore, \(x \geq 0\) is consistent with the equation.

Key Concepts

Understanding Absolute Value

The absolute value of a number measures its distance from zero on a number line. Because distance cannot be negative, absolute values are always non-negative numbers. For any real number \( x \), the absolute value is represented as \( |x| \). It is defined as follows:

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \).

This means, for example, that \( |3| = 3 \) and \( |-3| = 3 \).
When you take the absolute value of a number, you are essentially ignoring its sign and focusing only on its size. This is why the absolute value is crucial when solving equations, as it ensures that you consider all possible scenarios in the solution.

Exploring Square Roots

Square roots are mathematical operations that 'undo' the squaring of a number, essentially finding the original number that, when multiplied by itself, yields the given value. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \).

• The operation is denoted by the radical symbol \( \sqrt{} \).
• Square roots often deal with perfect squares, like 4, 9, or 16, where the result is an integer.
• Importantly, when you take the square root of \( x^2 \), you obtain \( |x| \). This is because squaring a number always results in a non-negative product, and the square root process must maintain non-negative results.

Recognizing this, you can solve equations involving square roots by considering the non-negative nature of the output, ensuring solutions align with the mathematical principle that square roots return non-negative values by default.

Importance of Non-Negative Numbers

Non-negative numbers include zero and all positive numbers. These numbers are represented as \( x \geq 0 \). In many mathematical contexts, particularly those involving square roots and absolute values, focus is often on non-negative numbers because they fit naturally within operations that produce non-negative results.

• Absolute values turn any number into its non-negative counterpart.
• Square roots, by definition, return only the non-negative root of a positive number.
• In solving equations like \( \sqrt{|x|^2} = x \), recognizing that the output (x) must be non-negative is key, as it aligns with the result of these operations.

Understanding these concepts helps provide clarity, especially in equations where ensuring non-negative results consistently leads to correct solutions.
Thus, when addressing such problems, focusing on non-negative potential solutions helps in maintaining mathematical consistency and correctness.