Question

In Problems 49-58, find the value of each expression if \(x=3\) and \(y=-2\). \(\frac{|y|}{y}\)

Short answer

-1

Step-by-step solution

Substitute the Given Values

Replace the variables with the given numbers. You are given that \(x = 3\) and \(y = -2\). So, substitute \(y\) with \(-2\) in the expression.

Evaluate the Absolute Value

Calculate the absolute value of \(y\). The absolute value of \(y\) is \(|y|\), which is \(|-2|\). This equals to 2 because the absolute value of a number is always positive.

Divide the Absolute Value by \(y\)

Now, divide \(|y|\) by \(y\). So, \(\frac{|y|}{y} = \frac{2}{-2}\).

Simplify the Fraction

Finally, simplify the fraction \(\frac{2}{-2}\). Dividing 2 by -2 gives -1.

Key concepts

absolute value

Absolute value is a concept that helps us understand the distance of a number from zero on the number line, regardless of its direction. It's represented by vertical bars, like this: \(|-|y||\). For example, consider \(|-2||\) and \(2||\). Though -2 and 2 are on opposite sides of zero on the number line, they are both 2 units away from zero.

variable substitution

Variable substitution is fundamental in algebra to solve equations and expressions with given values. For instance, if we are given \(y = -2||\), we directly replace 'y' in all expressions with -2. Hence, \(\frac{\|y\|}{y} ||\) becomes \(\frac{\|-2\|}{-2} ||\) upon substitution.

fraction simplification

Simplifying fractions involves reducing them to their simplest form. This means making the numerator and denominator as small as possible. Consider the expression \(\frac{|y|}{y} ||\). Substituting \(y||\) with -2, we get \(\frac{2}{-2} ||\). Here, the fraction simplifies to -1 because \(1||\) is the highest common factor of 2 and -2, enabling cancellation.

algebraic expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. For example, \(\frac{|y|}{y} ||\) is an algebraic expression where \(y||\) is a variable, \(|-|y||\) is the absolute value operation, and \(\frac{a}{b}||\) represents the division. Through substitution and simplification, we can evaluate these expressions.