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Chapter 12: Problem 256

Illustrate one (a) conditional inequality, (b) identity, and (c) inconsistent inequality.

Short Answer

Expert verified
(a) Conditional Inequality: \(x > 4\) (b) Identity: \(\sin^2(x) + \cos^2(x) = 1\) (c) Inconsistent Inequality: \(x - 2 > x + 1\)

Step by step solution

01

(a) Step 1: Choose a Conditional Inequality

Let's consider the following conditional inequality: \[2x - 3 > 5\]
02

(a) Step 2: Solve the Inequality

To solve the inequality, we need to isolate the variable x on one side: \[2x > 8\] Now, divide both sides by 2: \[x > 4\] This inequality means that the statement is true for any x such that \(x > 4\), which demonstrates a conditional inequality. #Phase 2: Illustrate an Identity#
03

(b) Step 1: Choose an Identity

Let's consider the following identity: \[\sin^2(x) + \cos^2(x) = 1\]
04

(b) Step 2: Verify the Identity

According to trigonometric properties, the sum of the squares of the sine and the cosine functions of the same angle will always equal 1, regardless of the angle's value. For example, let's take \(x = 45\degree\). \[\sin^2(45\degree) + \cos^2(45\degree) = (\frac{\sqrt{2}}{2})^2 + (\frac{\sqrt{2}}{2})^2 = \frac{1}{2} + \frac{1}{2} = 1\] This identity is true for all permissible values of x. #Phase 3: Illustrate an Inconsistent Inequality#
05

(c) Step 1: Choose an Inconsistent Inequality

Let's consider the following inconsistent inequality: \[x - 2 > x + 1\]
06

(c) Step 2: Attempt to Solve the Inequality

Try to isolate the variable x on one side: Subtract x from both sides: \[-2 > 1\] This statement is never true, which means there are no solutions for x; this demonstrates an inconsistent inequality.

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