Problem 67 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10 .\) See Example 6. $$ \frac{|x-(-10)|}{2 t} $$

Step-by-Step Solution

Answer

The simplified value of the expression is \( \frac{1}{4} \).

Step 1: Substitute the Values

Substitute the values of the variables into the expression. Given that \( x = -5 \), \( y = 4 \), and \( t = 10 \), substitute these values into the expression \( \frac{|x - (-10)|}{2t} \) to get \( \frac{|-5 - (-10)|}{2 \times 10} \).

Step 2: Simplify the Expression Inside the Absolute Value

Calculate the expression inside the absolute value: \( -5 - (-10) \) which simplifies to \( -5 + 10 = 5 \).

Step 3: Apply the Absolute Value

The absolute value of the simplified expression \( |5| \) is 5, as absolute value represents the distance from zero and is always non-negative.

Step 4: Substitute the Absolute Value Result and Simplify the Fraction

After finding the absolute value result, substitute it back into the expression: \( \frac{5}{2 \times 10} \).

Step 5: Simplify the Denominator

Calculate the denominator: \( 2 \times 10 = 20 \). Now, the expression simplifies to \( \frac{5}{20} \).

Step 6: Calculate the Final Answer

Simplify the fraction \( \frac{5}{20} \) to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 5. Thus, \( \frac{5}{20} = \frac{1}{4} \).

Key Concepts

Absolute Value

The absolute value, denoted by vertical bars such as \(|x|\), is an essential concept in mathematics. It measures the distance of a number from zero on the number line. \Absolute values are always non-negative, which means they can never be less than zero. For example, the absolute value of 5 and -5 is the same: \(|5| = 5\) and \(|-5| = 5\). \So, when you see an expression like \(|x - (-10)|\), you first calculate \(x - (-10)\) to get a number, say 5, and then take its distance from zero, which remains 5. \Understanding absolute values helps you solve situations where you need a consistent measure of distance, regardless of direction.

Substitution Method

The substitution method is a straightforward approach to solving expressions by replacing variables with their given numerical values. \In our exercise, we are given specific values for variables: \(x = -5\), \(y = 4\), and \(t = 10\). \To substitute, plug these values into the expression \(\frac{|x - (-10)|}{2t}\). \Start by replacing the variable \(x\) with \(-5\) and \(t\) with 10, leading to the expression \(\frac{|-5 - (-10)|}{2 \times 10}\). \This method simplifies problems by removing variables and transforming expressions into purely numerical forms.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that involves reducing expressions to their simplest form. \In the exercise, simplification occurs first by resolving the absolute value component, \(-5 - (-10)\), which becomes \(-5 + 10 = 5\). \Then, the expression \(\frac{|5|}{20}\) is further simplified by recognizing that the absolute value of 5 is 5 itself. \Simplifying supports clearer calculation and understanding by cutting down complicated terms, allowing you to focus on the core numerical operations.

Fractions

Fractions represent parts of a whole and are composed of a numerator and a denominator. \In the exercise, once the absolute value computation is complete, you work with the fraction \(\frac{5}{20}\). \Simplifying a fraction involves finding the greatest common divisor of the numerator and the denominator. Here, dividing both 5 and 20 by their greatest common divisor, 5, reduces the fraction to \(\frac{1}{4}\). \This process ensures fractions are presented in their simplest form, aiding in comparison and computation across different math problems.