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Chapter 1: Problem 67

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

Short Answer

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

Step by step solution

01

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 \( \frac{|-5 - (-10)|}{2 \times 10} \).
02

Simplify the Expression Inside the Absolute Value

Calculate the expression inside the absolute value: \( -5 - (-10) \) which simplifies to \( -5 + 10 = 5 \).
03

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.
04

Substitute the Absolute Value Result and Simplify the Fraction

After finding the absolute value result, substitute it back into the expression: \( \frac{5}{2 \times 10} \).
05

Simplify the Denominator

Calculate the denominator: \( 2 \times 10 = 20 \). Now, the expression simplifies to \( \frac{5}{20} \).
06

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, \( \frac{5}{20} = \frac{1}{4} \).

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Key Concepts

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

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.

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Most popular questions from this chapter

Fill in the table with the opposite (additive inverse), and the reciprocal (multiplicative inverse). Assume that the value of each expression is not 0 $$ \frac{1}{2 x} $$

Without calculating, determine whether each answer is positive or negative. Then use a calculator to find the exact difference. \(4.362-7.0086\)

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. The difference of sixteen and four is greater than ten.

Perform the following operations. Write answers in lowest terms. $$ 4 \frac{3}{7}+\frac{31}{7} $$

Perform the following operations. Write answers in lowest terms. $$ \frac{10}{3}-\frac{5}{21} $$
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