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Chapter 6: Problem 10

Is a conditional equation true for all real values in its domain?

Short Answer

Expert verified
No, a conditional equation is not true for all real values in its domain. It is only true for values that satisfy the condition of the equation.

Step by step solution

01

Understanding a conditional equation

A conditional equation is an equation that is true for certain values and false for others. This means an equation's truthiness depends on the values we substitute in place of the variables.
02

Understanding of Real Numbers and Domain

Real numbers are a set of numbers that includes both rational and irrational numbers. The domain of a function is the complete set of possible values of the independent variable. In our case, they are talking about the domain of all real numbers, which means all numbers that fall in the real number set.
03

Drawing a conclusion

With an understanding of both a conditional equation and the domain of real numbers, it can be concluded that a conditional equation is not necessarily true for all real values in its domain. It works for only certain values which satisfy the condition of the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Numbers
Rational numbers are numbers that can be expressed as a fraction or a ratio of two integers, where the denominator is not zero. This includes numbers like 1/2, 3, -5, or even 0.75. Here's what makes rational numbers special:
  • They can always be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \).
  • All integers are rational numbers because they can be expressed as the integer divided by 1.
  • They have either terminating or repeating decimal forms. For example, 0.75 is a terminating decimal, while 1/3 is a repeating decimal (0.333...).
Rational numbers are part of the set of real numbers, alongside irrational numbers.
Irrational Numbers
Irrational numbers are real numbers that cannot be written as a simple fraction or ratio. They have non-terminating and non-repeating decimal expansions. Unlike rational numbers, they cannot be expressed in the form \( \frac{p}{q} \). Here are some key aspects:
  • Famous examples include \( \pi \) and \( \sqrt{2} \), both of which have decimal forms that go on forever without repeating.
  • Irrational numbers are crucial to different areas in mathematics like geometry and calculus.
  • Even though they can't be written as simple fractions, they are critically important for describing and solving real-world problems.
Like rational numbers, irrational numbers are also part of the broader category of real numbers.
Domain of a Function
The domain of a function refers to all the possible input values (usually \( x \)) that can be used in the function without causing any undefined operations, like division by zero or square roots of negative numbers. Understanding the domain is important because it clarifies where the function is valid. Here鈥檚 how to identify the domain:
  • Look for any restrictions due to operations like division, taking roots, or logarithms. For example, \( \frac{1}{x} \) cannot have 0 as a part of its domain because division by zero is undefined.
  • In general polynomials, all real numbers are in the domain because they do not include operations that can become undefined.
Knowing the domain helps in understanding conditional equations, as it reveals the real values that can make the equation true within its context.

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Most popular questions from this chapter

Use the product-to-sum formulas to write the product as a sum or difference. $$3 \sin 2 \alpha \sin 3 \alpha$$

Determine whether the statement is true or false. Justify your answer. $$\sin \frac{x}{2}=-\sqrt{\frac{1-\cos x}{2}}, \quad \pi \leq x \leq 2 \pi$$

Find the solution(s) of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your results. $$2 \sin \left(x+\frac{\pi}{2}\right)+3 \tan (\pi-x)=0$$

The range of a projectile fired at an angle \(\theta\) with the horizontal and with an initial velocity of \(v_{0}\) feet per second is given by $$r=\frac{1}{32} v_{0}^{2} \sin 2 \theta$$ where \(r\) is measured in feet. An athlete throws a javelin at 75 feet per second. At what angle must the athlete throw the javelin so that the javelin travels 130 feet?

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin 5 \theta-\sin \theta$$
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