Problem 43 - A Graphical Approach to College Algebra

Question

Solve each equation and inequality. (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) All real x except x=2.

Step 1: Simplify the Numerator for both (a) and (b)

First, expand the expression in the numerator: \((x-2)(2) - (2x+1)(1)\). This simplifies to \[2x - 4 - (2x + 1) = 2x - 4 - 2x - 1 = -5.\] Thus, the numerator is constant (-5).

Step 2: Solve Equation (a)

For equation (a), set the simplified expression equal to zero:\[-5 = 0.\]Since this is false, the equation has no solution.

Step 3: Solve Inequality (b)

For inequality (b), determine when the simplified expression is less than zero:\[\frac{-5}{(x-2)^2} < 0.\]This is always false because \((x-2)^2\) is always positive and the numerator is negative, resulting in \(-5\) being divided by a positive number, which remains negative. Therefore, the inequality holds for all x except when it causes division by zero, i.e., \(x eq 2\).

Key Concepts

Equations and Inequalities

When dealing with equations, we seek to find the value of the variable that makes the equality true. Inequalities, on the other hand, determine the range of values that make one side greater or lesser than the other. In the presented problem, we explore both concepts through fractions set within an equation and inequality.
In part (a), the equation is simplified to \(\frac{-5}{(x-2)^2} = 0\). For an equation where a fraction equals zero, the numerator must also be zero. Here, \(-5 = 0\) is impossible, indicating no solution exists.
For part (b), we examine an inequality\(\frac{-5}{(x-2)^2} < 0\). Fractions being less than zero occur when the numerator and denominator have opposite signs. Since \(x-2)^2\) is always non-negative, the inequality \(-5 < 0\) holds under all circumstances, except when the denominator equals zero (i.e., \(x ≠ 2\)).

Simplification

Simplification is the process of rewriting expressions in a more concise or efficient form without changing their value. In the problem given, simplification was key to finding solutions quickly.
First, simplify the numerator of the expression by expanding and combining like terms. The expression \((x-2)(2) - (2x+1)(1)\) is expanded into \(2x - 4 - 2x - 1\). After simplifying, this results in \(-5\). By reducing it to a single constant, we alleviate further calculations, primarily focusing on the denominator for further evaluations.
Simplification thus serves as a foundational skill in algebra, aiding in the reduction of complex expressions to simpler forms that are easier to analyze.

Numerator and Denominator Analysis

Analyzing the numerator and denominator is crucial in understanding how a fraction behaves. The numerator determines the sign and overall value of the fraction when combined with the denominator's influence.
In our problem, the numerator \(-5\) is constant, simplifying our task as it never changes with different values of \(x\).
The denominator \((x-2)^2\) tells us how the expression behaves across various values of \(x\). A square function, like \((x-2)^2\), is always positive, except at \(x = 2\), where division by zero is undefined. As a result, the entire fraction remains negative, because a negative number divided by a positive one retains a negative sign.
Understanding how numerators and denominators interact clarifies when values are undefined and how they affect the solution set of equations and inequalities.