Problem 99 - Algebra and Trigonometry

Question

In Exercises \(98-101\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \(x+3,\) resulting in the equivalent inequality \(x-2<2(x+3)\)

Step-by-Step Solution

Answer

FiveThe given statement is false. The correct statement is: 'The inequality \(\frac{x-2}{x+3}<2\) can be solved by segregating cases depending on the sign of \(x+3\), leading to \(x - 2 < 2(x + 3)\) when \(x > -3\) and \(x - 2 > 2(x + 3)\) when \(x < -3\).'

Step 1: Validate the Given Statement

Firstly, check whether the given statement is true or not. For this, replace \(x\) with a test value, for example, 0 in both inequalities \(\frac{x-2}{x+3} < 2\) and \(x-2 < 2(x+3)\) and evaluate them.

Step 2: Determine the Correct Transformation

If the outcomes from Step 1 don't match, the given statement is false. In that case, identify the correct operation to maintain the equation's balance. Since the denominator may be negative, the inequality's direction may vary when multiplied by \(x+3\). Generally multiplying both sides by \(x+3\) won't result in the equivalent inequality, so the correct procedure will be to segregate cases based on the sign of \(x+3\).

Step 3: Correct the Statement and Generate a True Statement

To produce a true statement, these are the correct transformations: when \(x > -3, \frac{x-2}{x+3} < 2\) will transform to \(x - 2 < 2(x + 3)\) and when \(x < -3, \frac{x-2}{x+3} < 2\) will transform to \(x - 2 > 2(x + 3)\)

Key Concepts

Algebraic Expressions

Algebraic expressions are the heart of algebra and consist of numbers, variables, and operations combined to represent a mathematical relationship. In our exercise, the algebraic expressions are particularly embedded in the inequality \( \frac{x-2}{x+3} < 2 \). Here, the numerator and denominator are algebraic expressions, where \(x\) is the variable.
Representing real-world situations in mathematical terms often involves writing algebraic expressions.
These expressions can take many forms:

• Monomials: These consist of a single term, e.g., \( 5x \).
• Binomials: These involve two terms, e.g., \( 3x + 4 \).
• Polynomials: More complex combinations involving multiple terms, e.g., \( x^2 - 4x + 4 \).

In solving inequalities, understanding how to manipulate these expressions is essential. This often involves simplifying or transforming them through factorization, distribution, and applying algebraic operations.

Inequality Transformation

When dealing with inequalities, transforming them into an equivalent form is key to finding solutions. Inequalities are mathematical statements indicating that one side is either less than, greater than, or not equal to the other side. By carefully manipulating these statements, we can unveil the solution set.
In the exercise given, transforming \( \frac{x-2}{x+3} < 2 \) into another form involves strategic application of algebraic rules. Initially, one might think of multiplying both sides by \( x+3 \) to remove the fraction. However, when multiplying an inequality by an expression that includes a variable (like \( x+3 \)), the direction of the inequality might change based on if \( x+3 \) is positive, negative, or zero.
A helpful approach is to:

• Consider the critical points where the expression transitions from positive to negative, e.g., where \(x+3=0\).
• Divide the solution process into cases depending on the sign of the denominator.
• Account for every instance where the operation could impact the inequality's direction.

Transforming inequalities accurately leads to valid comparisons, helping to build a more reliable solution pathway.

Solving Inequalities

The process of solving inequalities involves finding the values of the variable that make the inequality true. With algebraic equations, the solutions are typically fixed numbers. However, in inequalities, the solutions form a range of values.
Our task is to solve the inequality \( \frac{x-2}{x+3} < 2 \). To do so:

• Identify critical points, such as \(x=-3\) where the denominator becomes zero, which must be considered separately as it can change the inequality's behavior.
• Break the problem into two cases: \( x > -3 \) and \( x < -3 \). These delineate different solution pathways since the sign of the denominator alters the inequality's direction.
• Solve each case separately, checking for consistency and validating these solutions within the original constraints of the initial inequality.

By effectively applying these steps, we reach a comprehensive understanding of the solutions' behavior over the problem domain. Solving inequalities equips us with insights that are crucial to both algebra and representing real-world constraints.