Problem 66
Question
Evaluate each algebraic expression for x = 2 and y = -5. $$\frac{|x|}{x}+\frac{|y|}{y}$$
Step-by-Step Solution
Verified Answer
The value of the given expression when \(x = 2\) and \(y = -5\) is \(0\).
1Step 1: Substituting the given values into the expression
Firstly, substitution of the given values: \(x = 2\) and \(y = -5\) into \(\frac{|x|}{x}+\frac{|y|}{y}\) is carried out. So, the expression becomes \(\frac{|2|}{2}+\frac{|-5|}{-5}\).
2Step 2: Evaluating the absolute values
In the next step, evaluate each absolute value in the expression. The absolute value of \(2\) is \(2\), and the absolute value of \(-5\) is \(5\). This changes the expression to \(\frac{2}{2}+\frac{5}{-5}\).
3Step 3: Simplifying the expression
Now, compute the fractions \(\frac{2}{2}\) and \(\frac{5}{-5}\), which are equal to \(1\) and \(-1\) respectively. So, the expression becomes \(1+ (-1)\).
4Step 4: Final Calculation
Finally, perform the operation \(1 - 1\), which results in \(0\).
Key Concepts
Absolute ValueSubstitution MethodSimplifying ExpressionsArithmetic Operations
Absolute Value
Understanding the absolute value concept is crucial for evaluating algebraic expressions that include this operation. Absolute value refers to the distance of a number from zero on a number line, irrespective of the direction. To put it simply, the absolute value of a number is always non-negative. It is denoted by two vertical bars surrounding the number, for example, the expression for the absolute value of x is written as \(|x|\).
When we apply the concept to the given exercise, the absolute value of 2 is 2 because it is two units away from 0 on the number line. Similarly, the absolute value of -5 is 5 because it is five units away from zero in the opposite direction. Thus, even though -5 is a negative number, its absolute value turns out to be positive as the distance is measured in positive units. The operation of absolute value transforms both positive and negative inputs into positive outputs, simplifying further calculations in the expression.
When we apply the concept to the given exercise, the absolute value of 2 is 2 because it is two units away from 0 on the number line. Similarly, the absolute value of -5 is 5 because it is five units away from zero in the opposite direction. Thus, even though -5 is a negative number, its absolute value turns out to be positive as the distance is measured in positive units. The operation of absolute value transforms both positive and negative inputs into positive outputs, simplifying further calculations in the expression.
Substitution Method
The substitution method is a fundamental technique used to evaluate algebraic expressions. It involves replacing the variables in an expression with their respective numerical values. The main idea is to simplify the expression to a point where computation can be done purely with numbers.
In our example, this method is initiated by substituting the values given for x and y into the original expression. Specifically, when we are given \(x = 2\) and \(y = -5\), these values are directly replaced in place of the variables. Thus, the algebraic expression \(\frac{|x|}{x} + \frac{|y|}{y}\) is transformed into \(\frac{|2|}{2} + \frac{|-5|}{-5}\) as the first step towards finding the value of the expression.
In our example, this method is initiated by substituting the values given for x and y into the original expression. Specifically, when we are given \(x = 2\) and \(y = -5\), these values are directly replaced in place of the variables. Thus, the algebraic expression \(\frac{|x|}{x} + \frac{|y|}{y}\) is transformed into \(\frac{|2|}{2} + \frac{|-5|}{-5}\) as the first step towards finding the value of the expression.
Simplifying Expressions
Simplifying expressions is about making an algebraic expression easier to understand and solve by performing a series of arithmetic steps. The goal is to rewrite the expression in the simplest form possible. This may include simplifying fractions, combining like terms, and applying basic arithmetic principles.
After substitution and evaluating the absolute values in our exercise, the expression \(\frac{2}{2} + \frac{5}{-5}\) needs to be simplified. This is done by dividing the numbers to simplify the fractions to their lowest terms. In this case, we get 1 and -1, since dividing a number by itself yields 1, and a positive number divided by its negative counterpart yields -1. After this step, the original complicated-looking expression is now reduced to a simple arithmetic problem of adding the numbers 1 and -1.
After substitution and evaluating the absolute values in our exercise, the expression \(\frac{2}{2} + \frac{5}{-5}\) needs to be simplified. This is done by dividing the numbers to simplify the fractions to their lowest terms. In this case, we get 1 and -1, since dividing a number by itself yields 1, and a positive number divided by its negative counterpart yields -1. After this step, the original complicated-looking expression is now reduced to a simple arithmetic problem of adding the numbers 1 and -1.
Arithmetic Operations
Arithmetic operations are the building blocks of mathematics. They include operations like addition, subtraction, multiplication, and division. Evaluating algebraic expressions often involves these operations to compute the final answer once the expressions have been simplified enough.
In the final step of the provided exercise, we conduct the simple arithmetic operation of adding 1 and -1 together. It is imperative to remember that when adding a positive number to its negative, the result is always zero because they cancel each other out. This is representative of finding a balance point or equilibrium on a number line. Thus, applying this operation to our simplified expression from the previous step gives us the final answer, which is 0.
In the final step of the provided exercise, we conduct the simple arithmetic operation of adding 1 and -1 together. It is imperative to remember that when adding a positive number to its negative, the result is always zero because they cancel each other out. This is representative of finding a balance point or equilibrium on a number line. Thus, applying this operation to our simplified expression from the previous step gives us the final answer, which is 0.
Other exercises in this chapter
Problem 66
Perform the indicated operations. Indicate the degree of the resulting polynomial. $$\left(5 x^{4} y^{2}+6 x^{3} y-7 y\right)-\left(3 x^{4} y^{2}-5 x^{3} y-6 y+
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Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$\sqrt[6]{\frac{1}{64}}$$
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Factor completely, or state that the polynomial is prime. $$4 x^{2}-4 x-24$$
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Write each number in decimal notation without the use of exponents. $$6 \times 10^{-4}$$
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