Problem 3 - Thomas Calculus

Question

If \(2 < x < 6,\) which of the following statements about \(x\) are necessarily true, and which are not necessarily true? $$ \begin{array}{ll}{\text { a. } 0 < x < 4} & {\text { b. } 0 < x-2 < 4} \\\ {\text { c. } 1 < \frac{x}{2} < 3} & {\text { d. } \frac{1}{6} < \frac{1}{x} < \frac{1}{2}} \\ {\text { e. } 1 < \frac{6}{x} < 3} & {\text { f. }|x-4| < 2} \\\ {\text { g. }-6 < -x < 2} & {\text { h. }-6 < -x < -2}\end{array} $$

Step-by-Step Solution

Answer

Statements a, b, c, d, e, f, and h are true; statement g is not necessarily true.

Step 1: Analyzing Statement A

For statement a, we check whether the inequality \(0 < x < 4\) is always true given \(2 < x < 6\). Since the range of \(x\) is from just above 2 to just below 4, \(0 < x < 4\) is true for this region, making statement a necessarily true.

Step 2: Analyzing Statement B

For statement b, \(0 < x-2 < 4\), we simplify the inequalities to \(2 < x < 6\). This covers the full range of \(2 < x < 6\) exactly, so statement b is necessarily true.

Step 3: Analyzing Statement C

For statement c, \(1 < \frac{x}{2} < 3\), we solve for \(x\) by multiplying through by 2: \(2 < x < 6\). This matches the range given for \(x\), so statement c is necessarily true.

Step 4: Analyzing Statement D

For statement d, \(\frac{1}{6} < \frac{1}{x} < \frac{1}{2}\), we analyze the reciprocal to create inequalities involving \(x\): \(2 < x < 6\). This is consistent with the initial condition, so statement d is necessarily true.

Step 5: Analyzing Statement E

For statement e, \(1 < \frac{6}{x} < 3\), solving for \(x\) involves two inequalities: \(x>2\) and \(x<6\). Both conditions satisfy the initial range \(2 < x < 6\), so statement e is necessarily true.

Step 6: Analyzing Statement F

For statement f, \(|x-4| < 2\), this means \(-2 < x-4 < 2\), which simplifies to \(2 < x < 6\). This matches the given condition, so statement f is necessarily true.

Step 7: Analyzing Statement G

For statement g, \(-6 < -x < 2\), we solve for \(x\) by multiplying through by -1, reversing the signs: \(-2 < x < 6\). Since the lower bound is incorrect, this does not match the original range \(2 < x < 6\), making statement g not necessarily true.

Step 8: Analyzing Statement H

For statement h, \(-6 < -x < -2\), solving for \(x\) again involves reversing inequalities: \(2 < x < 6\). This matches the original range for \(x\), so statement h is necessarily true.

Key Concepts

Understanding Algebra in Inequality Solving

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and expressions. It is fundamental in solving inequalities, where the goal is to find the range or set of values that make the inequality true. When working with inequalities, specifically the expression \[2 < x < 6\], it's crucial to remain mindful of the bounds of the inequality. Algebraic manipulation involves basic operations like addition, subtraction, multiplication, or division to isolate variables or simplify expressions.For example, when analyzing the statement \[0 < x - 2 < 4\], manipulating it to align with the original inequality \[2 < x < 6\] helps in determining if the statement is necessarily true. Algebra helps to maintain equality and correctness as we transform these expressions, ensuring that the manipulated inequality remains faithful to its original form.Moreover, multiplying or dividing all terms of an inequality by a negative number will reverse its signs, a key detail that provides careful insights in algebraic problem-solving.

Mastering Problem-Solving with Inequalities

Problem-solving in mathematics often involves determining the truthfulness of statements, based on given conditions or inequalities. It's about finding a logical sequence of steps that lead to a solution. Let's delve into how inequalities aid problem-solving.When we analyze a specific statement like\[1 < \frac{x}{2} < 3\], problem-solving involves translating it into terms consistent with our known inequality \[2 < x < 6\]. By multiplying through by 2, you align the expression to discover that the statement must be true. This step-by-step approach requires careful attention to operations and transformations.Another example is examining \[-6 < -x < 2\], where handling negatives involves reversing inequality signs. Problem-solving requires verifying each process against the initial condition, ensuring the derived expressions are accurately compared with what is known.In successful problem-solving, each step builds on the previous one. Start with what you know, apply logical deductions, and methodically reach the conclusion.

Leveraging Mathematical Logic in Inequality Analysis

Mathematical logic underpins the clarity and precision required in dealing with inequalities. It involves using logical reasoning to handle the structure of mathematical statements, ensuring each transformation and conclusion is consistent.Consider inequalities that deal with absolute values, such as:\[|x-4| < 2\].This expands into two separate inequalities: \[-2 < x - 4 < 2\], which simplifies to \[2 < x < 6\]. Mathematical logic helps break down the absolute value into a comprehensible expression reflecting the original inequality.Analyzing the reciprocal expressions, such as \[\frac{1}{6} < \frac{1}{x} < \frac{1}{2}\],requires understanding reciprocal shifts and reversing inequalities. Logic dictates that this must transform into the satisfaction of \[2 < x < 6\].Using mathematical logic, you assess each expression for consistency with the base inequality, ensuring every logical step adheres to the original bounds or refinement.