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Chapter 4: Problem 289

In the following exercises, find the difference. $$\frac{x}{19}-\frac{8}{19}$$

Short Answer

Expert verified
\[ \frac{x - 8}{19} \]

Step by step solution

01

Identify the common denominator

Both fractions \[ \frac{x}{19} \] and \[ \frac{8}{19} \] have the same denominator of 19.
02

Subtract the numerators

Since the denominators are the same, subtract the numerators: \[ \frac{x}{19} - \frac{8}{19} = \frac{x - 8}{19} \]

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Key Concepts

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

common denominator
When subtracting fractions, it is crucial to identify a common denominator. The denominator is the bottom number in a fraction. For example, in the fraction \( \frac{3}{4} \), 4 is the denominator. Having a common denominator means that both fractions you want to subtract share the same denominator. If they do, you can proceed directly to subtracting the numerators. In our example, both fractions, \( \frac{x}{19} \) and \( \frac{8}{19} \) already have 19 as their common denominator, simplifying the subtraction process.
subtracting numerators
Once you have confirmed that the fractions have a common denominator, you can subtract the numerators. The numerator is the top part of the fraction. For instance, in \( \frac{7}{9} \), 7 is the numerator. To subtract fractions, you only need to subtract the numerators and retain the common denominator. In our case, subtract the 8 from x:

  • For \( \frac{x}{19} - \frac{8}{19} \), subtract the numerators: x - 8
  • Keep the common denominator unchanged: 19
This gives you a result of \( \frac{x - 8}{19} \).
algebraic expression
When working with fractions that involve variables, you often deal with algebraic expressions. An algebraic expression contains numbers and letters (variables) combined with operators like addition and subtraction. In the given exercise, the variable is x.
Algebraic expressions follow the same rules for fractions as numerical values do. You still need a common denominator to perform subtraction or addition. In this example, because both fractions \( \frac{x}{19} \) and \( \frac{8}{19} \) share the denominator of 19, we straightforwardly subtract the numerators:
  • Subtract 8 from x directly
  • Retain the denominator

This ensures your result is in a simplified form: \( \frac{x - 8}{19} \).

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Most popular questions from this chapter

In the following exercises, simplify. $$\left(\frac{5}{9}+\frac{1}{6}\right) \div\left(\frac{2}{3}-\frac{1}{2}\right)$$

In the following exercises, simplify. $$ \frac{\frac{p}{2}}{\frac{q}{5}} $$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. 2\(x^{2} y^{3}\) when \(x=-\frac{2}{3}\) and \(y=-\frac{1}{2}\)

In the following exercises, perform the indicated operation. $$ 3 \frac{5}{8}+3 \frac{7}{8} $$

In the following exercises, change to equivalent fractions using the given LCD. $$\frac{1}{3} \text { and } \frac{1}{5}, \quad L C D=15$$
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