Problem 46 - A Graphical Approach to College Algebra

Question

Solve each equation and inequality. (a) \(\frac{\left(x^{2}-1\right)(3)-(3 x-1)(2 x)}{\left(x^{2}-1\right)^{2}}=0\) (b) \(\frac{\left(x^{2}-1\right)(3)-(3 x-1)(2 x)}{\left(x^{2}-1\right)^{2}} \leq 0\)

Step-by-Step Solution

Answer

(a) No real solution. (b) Solution set is all real numbers except \(x = \pm 1\).

Step 1: Simplify the Numerator

First, focus on simplifying the numerator \((x^{2}-1)(3)-(3x-1)(2x)\). Distribute each term: \(3(x^2-1) = 3x^2 - 3\) and \(- (3x-1)(2x) = -6x^2 + 2x\). Combine like terms to get \(-3x^2 + 2x - 3\).

Step 2: Solve the Equation

Set the simplified numerator to 0: \(-3x^2 + 2x - 3 = 0\). Factor or use the quadratic formula to find the roots. Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = -3\), \(b = 2\), and \(c = -3\). The discriminant is \(2^2 - 4(-3)(-3) = 4 - 36 = -32\). Since the discriminant is negative, there are no real solutions.

Step 3: Analyze the Inequality

Consider the inequality \(\frac{-3x^2 + 2x - 3}{(x^2-1)^2} \leq 0\). Since the numerator is \(-3x^2 + 2x - 3\) which has no real roots, we know its sign remains constant across all real numbers. Evaluate the sign of \(-3x^2 + 2x - 3\) at a test point (e.g., \(x = 0\)). \(-3(0)^2 + 2(0) - 3 = -3\), thus the numerator is negative across all real numbers. The denominator \((x^2-1)^2\) is always positive except at \(x = \pm 1\), where it is zero.

Step 4: Determine Solution to the Inequality

The inequality involves a negative numerator over an always positive denominator (except \(x = \pm 1\)), implying that \(\frac{-3x^2 + 2x - 3}{(x^2-1)^2} < 0\) for all \(x eq \pm 1\). Therefore, the solution set is all real numbers except \(x = \pm 1\) because at these points, the expression is undefined.

Key Concepts

Quadratic Equation

Quadratic equations are a fundamental aspect of algebra, represented in the general form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The solution to a quadratic equation can be found using several methods, such as factoring, completing the square, or the quadratic formula.

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), allows us to find the roots of any quadratic equation. The term under the square root, \(b^2 - 4ac\), is known as the discriminant.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant equals zero, there is exactly one real root, or the roots are repeated.
• If the discriminant is negative, as in our exercise, there are no real roots, only complex ones.

Understanding how to work with the discriminant is crucial, as it determines the nature of the solutions to the quadratic equation.

Numerator Simplification

Numerator simplification is an essential step when dealing with complex algebraic expressions, especially within rational expressions and equations. Simplifying the numerator involves several sub-steps like distributing terms and combining like terms.

In the given exercise, the numerator is \((x^2-1)(3)-(3x-1)(2x)\). To simplify this, follow these steps:

• Distribute the terms: \(3(x^2 - 1) = 3x^2 - 3\) and then \(-(3x - 1)(2x) = -6x^2 + 2x\).
• Combine like terms to simplify: \(-3x^2 + 2x - 3\).

Once the numerator is simplified, it becomes much easier to handle the entire rational expression, enabling a more straightforward approach to solving equations or inequalities involving it.

Rational Expression

A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding and solving problems involving rational expressions, especially inequalities, requires careful analysis of the sign of the numerator and the properties of the denominator.

For example, in our exercise, the rational expression is \(\frac{-3x^2 + 2x - 3}{(x^2-1)^2}\). Solving inequalities with rational expressions typically involves assessing where the expression is undefined and determining the signs of the numerator and denominator.

• The numerator, \(-3x^2 + 2x - 3\), does not have real roots, so it remains negative for all real numbers.
• The denominator \((x^2-1)^2\) becomes zero at \(x = \pm 1\), where the expression is undefined.

Ultimately, the inequality \(\frac{-3x^2 + 2x - 3}{(x^2-1)^2} \leq 0\) is resolved by focusing on the ranges where both the numerator and denominator help satisfy the inequality. This analysis reveals that the expression is negative for all real numbers, except where the denominator equals zero, leading to solutions of all real numbers except \(x = \pm 1\).