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Chapter 0: Problem 65

In Exercises \(61-66,\) evaluate each algebraic expression for \(x=2\) and \(y=-5\) $$\frac{y}{|y|}$$

Short Answer

Expert verified
-1

Step by step solution

01

Substitute the Value of y

Initially, replace y in the expression \(\frac{y}{|y|}\) with its given value, which is -5. We now have \(\frac{-5}{|-5|}\)
02

Calculate Absolute Value

Next step is to calculate the absolute value of -5 denoted by |-5|. The absolute value of a number is the distance of that number from zero on a number line, regardless of direction. As distance is never negative, absolute values are always nonnegative. So, |-5| equals to 5
03

Perform Division Operation

Following this, perform the division \(\frac{-5}{5}\) to finalize the result. We obtain -1

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Key Concepts

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

Understanding Absolute Value
The concept of absolute value is fundamental in mathematics, particularly when dealing with algebraic expressions. Absolute value refers to the non-negative value of a number regardless of its sign. Imagine the number line where zero is the central point. The absolute value is how far a number is from zero on this line.
  • If the number is positive, its absolute value is simply the number itself.
  • If the number is negative, its absolute value is the positive form of the number.

For example, the absolute value of -5 is 5, represented as \(|-5| = 5\).
The importance of this function is that it always yields a non-negative number, making it useful in scenarios where negative numbers cannot be applied or interpreted.
The Role of Substitution
Substitution is a straightforward, yet essential operation in algebra. It involves replacing a variable in an expression with a given value, allowing you to simplify or solve the expression.
When you substitute a value into an algebraic expression, you must follow it accurately:
  • Identify the variable to be replaced, which in many problems is given by an equation or set of variables.
  • Replace the variable with its given numerical value.

For instance, in the expression \(\frac{y}{|y|}\), you substitute \(-5\) for \(y\), resulting in the simplified expression \(\frac{-5}{|-5|}\). This substitution shows the direct characteristic of \(y\). It sets the stage for further calculations, such as finding the absolute value and performing division.
Mastering Division Operation
Division is a basic arithmetic operation used frequently in algebra. It involves splitting a given quantity into equal parts. Division plays a critical role in simplifying algebraic expressions, especially when paired with other operations like absolute value.
Here's a simple way it works:
  • The fraction's numerator (top number) is divided by its denominator (bottom number).
  • The result is the quotient, which can be an integer, a fraction, or even a decimal if division doesn鈥檛 resolve evenly.

In the context of our example problem, after substituting the value of \(y\) and computing the absolute value, the expression becomes \(-5 \/ 5\). Here, division yields \(-1\) since dividing a negative numerator by a positive denominator results in a negative quotient. This operation clarifies how quantities compare relatively and completes the simplification of the expression.

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Most popular questions from this chapter

What is a compound inequality and how is it solved?

Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without discount passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$\frac{x^{2}+6 x+5}{x^{2}-25}$$
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