Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (a) \(\frac{(x-2)(2)-(2 x+1)(1)}{(x-2)^{2}}=0\) (b) \(\frac{(x-2)(2)-(2 x+1)(1)}{(x-2)^{2}}<0\)

Step-by-Step Solution

Answer

(a) No solution. (b) Solution is all real numbers except \(x = 2\).

Step 1: Simplify the Numerator

For both parts, start by simplifying the numerator \((x-2)(2)-(2x+1)(1)\). Distribute the constants in each term: \(2(x-2) - (2x + 1)\) becomes \(2x - 4 - 2x - 1 = -5\).

Step 2: Equation Component for Part (a)

Set the simplified numerator equal to zero: \[-5 = 0\].This equation is false, suggesting there is no solution for Part (a).

Step 3: Factor Inequality for Part (b)

Analyze the inequality \[-\frac{5}{(x-2)^2} < 0\].Note that \((x-2)^2\) is always non-negative for all \(x\). Since \(-5\) is negative, multiplying by a square (non-negative) keeps it negative.

Step 4: Determine Solution for Inequality

The fraction \(-\frac{5}{(x-2)^2}\) will always be negative for all real \(x\) except \(x = 2\) where the denominator is zero. Hence, the solution is all real numbers except \(x = 2\).

Step 5: Additional Clarification Step

Make sure not to include \(x = 2\) in the solution for the inequality, because division by zero is undefined.

Key Concepts

Equations

Let's dive into the concept of equations in calculus. An equation is a statement that asserts the equality of two expressions. It's like saying two things are the same in mathematical terms. In our specific exercise, we encountered an equation involving a fraction with variables in both the numerator and the denominator. The main goal when solving an equation like this is to first simplify it.

• Simplifying involves distributing and combining like terms to make the equation more manageable.
• Setting the simplified form equal to zero helps to find the potential solutions (if any exist!).

In this case, we ended up with \[-5 = 0\], which is a false statement. This indicates there's no solution for the equation part of this problem. When an equation cannot satisfy the conditions set (like in this example), it means there aren't any values of \(x\) that would make the equation true. This is a crucial concept to understand: not all equations will have solutions.

Inequalities

Now onto inequalities, which are just as important in calculus as equations. An inequality shows the relationship between two expressions that are not equal. It uses symbols like <, >, ≤, or ≥. Our inequality was \[\frac{(x-2)(2)-(2x+1)(1)}{(x-2)^{2}}<0\].

• The first step is similar to that of an equation: simplify the expression by distributing and simplifying the terms.
• Then, analyze the inequality. This involves understanding the behavior of the numerator and the denominator separately. In our problem, the simplified form became \[-\frac{5}{(x-2)^2} < 0\].
• This shows us that the entire expression's value depends on the denominator, \((x-2)^2\), which is squared and always non-negative except at \(x=2\), where it is undefined.

Inequalities often have many solutions, or solution intervals. In the case of this exercise, the expression is negative for all real numbers other than \(x = 2\). It's important to exclude \(x = 2\) because it makes the denominator zero, leading to an undefined expression. Visualizing inequalities graphically can often help students better understand which areas or values satisfy the inequality.

Step-by-Step Solutions

Finally, let's talk about step-by-step solutions. Providing these detailed explanations is incredibly helpful for learning calculus, especially when managing complex expressions.Clearly outlined steps guide you on how to approach, simplify, and solve different math problems such as equations and inequalities in calculus.

• Keep your work organized so you don't miss small details that could affect the overall solution.
• `Simplify the expressions`,`set the equation equal to zero` for finding solutions, or `analyze inequalities` by considering the signs and values of both the numerator and denominator.

In both the equation and inequality, we followed distinct steps:

• Simplifying the expression first by distributing and combining like terms.
• Then, setting it equal to zero for the equation or analyzing sign changes in the inequality.

This structured approach allows you to tackle math problems confidently and efficiently, ensuring you hone your problem-solving skills and deepen your understanding of calculus fundamentals. Remember to check each step along the way for accuracy and clarity.