Question

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

Short answer

1

Step-by-step solution

Understand the Absolute Value

The expression involves the absolute value of x, which is denoted by \(|x|\). By definition, the absolute value of a number is its distance from zero on a number line. It's always non-negative. For any real number \(x\), \(|x| = x\) if \(x \geq\ 0\) and \(|x| = -x\) if \(x < 0\).

Substitute Given Value

Substitute the given value \(x = 3\) into the expression \(|x|\). Since \(x = 3\) is a positive number, \(|3| = 3\).

Complete the Expression

Now, place the absolute value into the original fraction. The expression \(|x|/x\) becomes \(\frac{3}{3}\).

Simplify the Fraction

Simplify \(\frac{3}{3} = 1\).

Key concepts

Absolute Value

The absolute value of a number, represented as \(|x|\), is a fundamental concept in mathematics that indicates the distance of a number from zero on the number line. This distance is always a non-negative value. Essentially:

• If the number is positive or zero (\(x \geq 0\)), then \(|x| = x\).
• If the number is negative (\(x < 0\)), then \(|x| = -x\).

Let's apply this to our problem. Since our given value of \(x\) is 3 (a positive number), calculating the absolute value is straightforward:
\[|3| = 3\]This means that 3 is 3 units away from zero on the number line, and that is the value we use.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and the denominator are as small as possible and no common factors exist between them apart from 1. To simplify, you may use the following steps:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCF.

In our problem, after substituting and calculating the absolute value of \(x\), we had the fraction \(\frac{|x|}{x}\) which became \(\frac{3}{3}\). Since both the numerator (3) and the denominator (3) are the same, we simplify it by dividing both by their GCF, which is 3, thus:
\[\frac{3}{3} = 1\]}

Substitution in Algebra

Substitution in algebra is a technique where we replace variables with their given numerical values to simplify and solve expressions or equations. This technique is essential in verifying and understanding algebraic expressions. Here’s how you can perform substitution step-by-step:

• Identify the variables in your expression or equation.
• Replace each variable with the value provided for it.
• Perform any necessary arithmetic operations to simplify the expression.

In our example, the given value was \(x = 3\). We substituted \(3\) for \(x\) in the expression \(\frac{|x|}{x}\). First, we calculated the absolute value of \(3\), which is \(3\). Then our expression became \(\frac{3}{3}\), which we simplified to 1. Hence, by using substitution, we break down complex expressions into simpler and more manageable parts.