Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (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

To solve: set the numerator to zero and solve the resulting quadratic equation.

Step 1: Understand the Equation

We need to solve \( \frac{\left(x^{2}-1\right)(3)-(3 x-1)(2 x)}{\left(x^{2}-1\right)^{2}}=0 \) by finding the values of \( x \) that make the numerator zero, since for a fraction to equal zero, its numerator must be zero.

Step 2: Simplify the Numerator

Expand and simplify the numerator: \((3)(x^2-1) - (3x-1)(2x) = 3x^2 - 3 - (6x^2 - 2x) = 3x^2 - 3 - 6x^2 + 2x = -3x^2 + 2x - 3\).

Key Concepts

Equations

In calculus, equations form the backbone for many concepts and calculations. An equation represents a statement of equality between two expressions. When solving equations, especially in the context of calculus, the goal is to find the variable's value that makes the equation true. This might involve simplifying expressions, factoring, or sometimes applying more advanced techniques like derivatives or integrals.

For example, to solve an equation like \( \frac{(x^2-1)(3)-(3x-1)(2x)}{(x^2-1)^2}=0 \), you first need to focus on the denominator and numerator. Here, the fraction equals zero when its numerator equals zero as long as the denominator is not zero. Solving this kind of problem typically involves steps such as expanding and simplifying the expression.

Inequalities

Inequalities express a broader range of solutions compared to equations. In inequalities, you might be looking for a range of values rather than a single solution. The inequality \( \frac{(x^2-1)(3)-(3x-1)(2x)}{(x^2-1)^2} \leq 0 \) means that the expression is less than or equal to zero. Inequalities are solved in a similar fashion to equations; however, one needs to pay careful attention to the direction of the inequality.

Typically, you will solve the inequality by first addressing its expression, just like with equations, by simplifying the numerator. Once simplified, the inequality can be analyzed to identify intervals of \( x \) that satisfy the condition. Keep in mind important properties such as when multiplying or dividing by a negative number, the inequality flips direction.

Numerator simplification

Numerator simplification is a vital step in solving equations and inequalities involving fractions. In the given problem, the numerator is \( (3)(x^2-1) - (3x-1)(2x) \). By expanding these terms, we get \( 3x^2 - 3 \) and \( - (6x^2 - 2x) \). Simplification involves combining like terms to reduce the expression to \( -3x^2 + 2x - 3 \).

Detailed steps in simplification include:

• Multiply out each term separately.
• Carefully distribute negative signs through parentheses.
• Combine like terms, essentially gathering all \( x^2 \), \( x \), and constant terms together.

Doing this correctly allows you to focus on solving the equation or inequality based on the simpler form, making further calculations more straightforward and less error-prone.