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Chapter 1: Problem 76

Simplify each expression. See Examples I through 11 . $$ \frac{|-14|-|2-7|}{-15} $$

Short Answer

Expert verified
The simplified expression is -\(\frac{3}{5}\).

Step by step solution

01

Simplify Inside Absolute Values

First, let's simplify the expression inside the absolute values. We have two absolute values: 1. \(|-14|\) which simplifies to 14 because the absolute value of any number is the distance from zero, and it's always positive.2. \(|2-7|\) simplifies to \(|-5|\), which is 5.
02

Substitute Simplified Values Back

Replace the absolute values in the original expression with the simplified numbers. The expression becomes:\[\frac{14 - 5}{-15}\]
03

Subtract the Numerator

Perform the subtraction in the numerator. Here, subtract 5 from 14 to get 9. The expression is now:\[\frac{9}{-15}\]
04

Simplify the Fraction

Now simplify the fraction \(\frac{9}{-15}\). Both the numerator and the denominator can be divided by 3. Dividing 9 by 3 gives 3, and dividing -15 by 3 gives -5. Thus, the expression simplifies to:\[\frac{3}{-5}\]This can also be written as -\(\frac{3}{5}\).

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Key Concepts

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

Absolute Value
Understanding the concept of absolute value is about recognizing it as the non-negative distance a number has from zero on a number line. For example, the absolute value of \(-14\) is written as \(|-14|\), which equals 14. This is because no matter whether a number is negative or positive, its distance from zero is always considered positive.

In more literal terms, if you drop the sign of a number, what remains is its absolute value:
  • The absolute value of \(-14\) is 14, because you ignore the negative sign.
  • Similarly, the absolute value of \(2 - 7\) simplifies to \(-5\), and is \(|-5| = 5\).
Remember, when working with expressions involving absolute values, always resolve these first to continue properly with subsequent operations.
Numerator Subtraction
Numerator subtraction involves handling the operation of subtracting numbers at the top of a fraction. In our exercise, once the absolute values are resolved, the expression becomes \(\frac{14 - 5}{-15}\).

The next step is to address this subtraction in the numerator:

  • When you have \(14 - 5\): Simply take 5 away from 14, resulting in 9.
This process replaces the original numerator, leaving you with a new fraction \(\frac{9}{-15}\). Subtracting numbers directly simplifies the number in the numerator, a crucial step before further simplifying the fraction as a whole.
Fraction Simplification
Fraction simplification involves breaking down a fraction to its simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD).

Initially, we had the fraction \(\frac{9}{-15}\). Notice both 9 and -15 can be divided by 3, which is their GCD.

Here's what to do:
  • Divide both 9 and -15 by 3: \( \frac{9}{3} = 3 \)
  • Also, \(\frac{-15}{3} = -5\)
Thus, the fraction simplifies to \(\frac{3}{-5}\), which is equivalent to -\(\frac{3}{5}\).
Fraction simplification ensures you're working with the expression in its simplest (most reduced) form, which is often much easier to use and understand, especially for further calculations.

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Most popular questions from this chapter

Complete the statement to illustrate the given property. \(4(x+3)=\)_____Distributive property.

Use the distributive property to find the product. See Example II. $$ -10(2 a-3 b-4) $$

Write each of the following as an algebraic expression. See Examples 12 and 13. The cost of \(y\) books if each book costs \(\$ 35.61\)

Simplify each expression. See Examples I through 11 . $$ \frac{|3-9|-|-5|}{-3} $$

Evaluate each expression when \(x=9\) and \(y=-2 .\) See Example 12 $$ \frac{\sqrt{x}}{y}-\frac{y}{x} $$
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