Problem 96 - Algebra for College Students

Question

Which of the following equations is not an identity? Explain. a) \(\frac{x^{2}-1}{2} \cdot \frac{2}{x-1}=x+1\) b) \(\frac{x-1}{x^{2}-1}=x+1\) c) \(x^{2}-1=(x-1)(x+1)\) d) \(\frac{1}{x^{2}-1} \div \frac{1}{x+1}=\frac{1}{x-1}\)

Step-by-Step Solution

Answer

Equation b is not an identity because it does not hold true for all permissible values of x.

Step 1: Understand What an Identity Is

An identity is an equation that is true for all values of the variable for which both sides of the equation are defined. We need to check each equation for equality over the range of permissible values for x.

Step 2: Analyze Equation a

\(\frac{x^{2}-1}{2} \cdot \frac{2}{x-1} = x+1\) First, simplify the left-hand side: \( \frac{(x^{2}-1)\cdot 2}{2 \cdot (x-1)} = x+1\) Simplifying further we get: \( \frac{(x+1)(x-1)}{(x-1)} = x+1 \) Provided \(x eq 1\): \(x+1 = x+1\), which is always true.

Step 3: Analyze Equation b

\(\frac{x-1}{x^{2}-1} = x+1\) First, expand the denominator: \(x^{2}-1 = (x-1)(x+1)\) Then simplify the left-hand side: \(\frac{x-1}{(x-1)(x+1)} = x+1 \) Provided \(x eq \pm 1\): \(\frac{1}{x+1} = x+1 \). This is not true for all permissible values of \(x\), so this is not an identity.

Step 4: Analyze Equation c

\(x^{2}-1 = (x-1)(x+1)\) Expand the right-hand side: \((x-1)(x+1) = x^{2} - 1\) This equation is always true for all \(x\), so it is an identity.

Step 5: Analyze Equation d

\(\frac{1}{x^{2}-1} \div \frac{1}{x+1} = \frac{1}{x-1}\) First, rewrite the division as multiplication by the reciprocal: \(\frac{1}{x^{2}-1} \cdot \frac{x+1}{1} = \frac{1}{x-1}\) Simplify the left-hand side by expanding \(x^{2}-1\): \(\frac{1}{(x-1)(x+1)} \cdot (x+1) = \frac{1}{x-1}\) Provided \(x eq \pm 1\): \(\frac{1}{x-1} = \frac{1}{x-1}\), which is always true.

Key Concepts

equation analysis

When tackling algebraic problems, it's essential to analyze each equation thoroughly. Equation analysis involves dissecting an equation to understand its structure and to identify values for which it holds true. We check each equation's validity by simplifying both sides and testing various values of the variable, usually denoted as x. For instance, let's consider the equation \(\frac{x^2 - 1}{2} \times \frac{2}{x - 1} = x + 1\). To analyze this, we simplify the left-hand side: \(\frac{(x^2 - 1) \times 2}{2 \times (x - 1)} = x + 1\). Simplifying further, we get \(\frac{(x + 1)(x - 1)}{x - 1} = x + 1\). This equation holds true for all x ≠ 1, confirming that it is an identity in terms of equation analysis.

simplifying expressions

Simplifying expressions is a key skill in algebra that helps verify the truth of given equations. To simplify an algebraic expression, we factorize or combine like terms to make the expression easier to work with. Let's consider the equation \(\frac{x-1}{x^2 - 1} = x + 1\). First, we expand the denominator: \((x^2 - 1) = (x - 1)(x + 1)\) and then the left-hand side becomes: \(\frac{x - 1}{(x - 1)(x + 1)}\). Simplifying this, we get \(\frac{1}{x + 1} eq x + 1\) for all permissible values of x. As such, simplifying this expression reveals that it is not an identity.

identities in algebra

In algebra, an identity is an equation that holds true for all permissible values of the variable. Unlike conditional equations, identities are universally valid and help simplify complex problem solving. For example, the equation \(x^2 - 1 = (x - 1)(x + 1)\) is an identity. To verify, we expand the right-hand side: \((x - 1)(x + 1) = x^2 - 1\). This simplification shows the equation holds for all values of x. Identifying algebraic identities is crucial for further simplifying and solving algebraic problems efficiently.

variable constraints

Variable constraints refer to the limitations on the values that a variable can take in order to ensure the equation is defined and potentially true. These constraints are critical when validating an identity. For instance, in the equation \(\frac{1}{x^2 - 1} \times \frac{x + 1}{1} = \frac{1}{x - 1}\), we first rewrite the division as multiplication and simplify: \(\frac{1}{(x^2 - 1)} \times (x + 1) = \frac{1}{x - 1}\). For simplification, suppose \(x^2 - 1 = (x - 1)(x + 1)\), then we have: \(\frac{1}{(x - 1)(x + 1)} \times (x + 1) = \frac{1}{x - 1}\). The constraint here is that x cannot be ±1, as this would make the denominator zero, invalidating the equation. Such constraints are essential for determining whether an equation serves as a valid identity.