Problem 41 - Algebra 2

Question

REASONING Determine whether the equation \(\frac{\sqrt{\left(x^{2}\right)}^{2}}{-x}=x\) is sometimes, always, or never true when \(x\) is a real number. Explain your reasoning.

Step-by-Step Solution

Answer

The equation is sometimes true, specifically when \(x = 0\).

Step 1: Simplify the Left Side

First, simplify the expression on the left side of the equation \(\frac{\sqrt{(x^{2})}^{2}}{-x}\). Notice that \(\sqrt{(x^2)}\) simplifies to \(|x|\), which means that the square root of \(x^2\) is the absolute value of \(x\). Thus, \(\sqrt{(x^2)}^2 = x^2\).

Step 2: Simplify Further Using Properties of Absolute Value

Now take \(x^2\) and simplify to \(\frac{x^2}{-x}\). Since \(x^2 = x \cdot x\), we have \(\frac{x \cdot x}{-x} = \frac{x}{-1} \cdot x = -x\). However, recall that for \(x \lt 0\), \(\frac{x^2}{-x} = -x\), and for \(x \gt 0\), \(\frac{x^2}{-x} = -x\).

Step 3: Compare Both Sides of the Equation

Now the original equation \(\frac{\sqrt{(x^2)}^2}{-x} = x\) simplifies to \(-x = x\). This implies \(-x = x\), which further implies that \(x = 0\) when both sides are equal.

Step 4: Determine the Conditions for the Equation

Evaluate the resulting equation \(-x = x\). This equation is only true when \(x = 0\). For all other real numbers, \(-x eq x\), making the equation false. Therefore, the equation is only sometimes true, based on whether \(x = 0\).

Key Concepts

Real Numbers

Real numbers are all the numbers you can find on the number line. These include:

• Positive numbers, like 1, 2, 3.
• Negative numbers, like -1, -2, -3.
• Zero, which is neither positive nor negative.
• Fractions and decimals, such as \(\frac{1}{2}\) and 3.14.

Real numbers are very useful in algebra, helping us solve equations just like in the exercise we discussed. They allow for a broad range of operations, including addition, subtraction, multiplication, and division (except by zero). Even though zero isn’t positive or negative, it plays a special role because it separates these two groups of numbers.
A critical point in our example is understanding that the equation's truthfulness can change depending on which real number \(x\) we use. For instance, when \(x = 0\), the equation behaves differently compared to when \(x\) is positive or negative, highlighting the unique properties of real numbers.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, without considering direction. For example, both -3 and 3 have an absolute value of 3. The absolute value of a real number \(x\) is denoted as \(|x|\).
This concept is vital in our exercise because when you simplify \(\sqrt{(x^2)}\), it becomes \(|x|\). This transformation to absolute value helps simplify the left side of the given algebraic equation.

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

Understanding absolute value is crucial for correctly interpreting the solutions to equations, as it ensures that our solutions are interpreted in terms of their magnitude, not their direction. Hence why it's important to remember that \(\sqrt{(x^2)} = |x|\) manages both positive and negative numbers correctly.

Simplifying Expressions

Simplifying expressions in algebra is all about making them easier to work with. This usually involves reducing fractions, combining like terms, or rewriting expressions using known identities, such as \(a^2 - b^2 = (a-b)(a+b)\).
In our specific problem, we simplified \(\frac{\sqrt{(x^2)}^2}{-x}\) to \(-x\) by applying the property that \(\sqrt{(x^2)}\) equals \(|x|\). By acknowledging that substituting back the simplified parts of the expression helps to show whether the left side equals the right side, we reach the core of why \(-x = x\) only holds true when \(x = 0\).
This process is important not just for solving equations but also for making sure they properly reflect the mathematical relationships we want to communicate.

Properties of Equality

The properties of equality are foundational rules used in solving algebraic equations. They help us manipulate equations while keeping them balanced. These properties include:

• Reflexive Property: Any number is equal to itself. For example, \(a = a\).
• Symmetric Property: If \(a = b\), then \(b = a\).
• Transitive Property: If \(a = b\) and \(b = c\), then \(a = c\).
• Additive Property: If \(a = b\), then \(a + c = b + c\).
• Multiplicative Property: If \(a = b\), then \(ac = bc\).

Using these properties correctly ensures that when we manipulate equations, such as going from \(-x = x\) to finding that \(x = 0\), we maintain valid transformations. In our example problem, we critically used these properties to deduce that the equation \(-x = x\) only holds valid when \(x = 0\). Recognizing which property to use at each step is vital for solving equations efficiently and accurately.