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Chapter 7: Problem 59

Add or subtract. Write answer in lowest terms. \(\frac{3}{4 p-5}+\frac{9}{5-4 p}\)

Short Answer

Expert verified
\(\frac{-6}{4p-5}\)

Step by step solution

01

Simplify the Denominators

Observe that the denominators in the two fractions are similar but with opposite signs: \(4p - 5\) and \(5 - 4p\). Rewrite one of the denominators to have the same sign for easier addition: \(\frac{9}{5-4p} = \frac{9}{-(4p-5)} = -\frac{9}{4p-5}\).
02

Adjust the Expression

After rewriting, the expression becomes: \(\frac{3}{4p-5} + \left(-\frac{9}{4p-5}\right)\).
03

Combine Like Terms

Now that the denominators are the same, combine the numerators: \(\frac{3 - 9}{4p-5} = \frac{-6}{4p-5}\).
04

Simplify the Fraction

Simplify the fraction if possible. In this case, the numerator and denominator have no common factors other than 1, so the fraction is already in its simplest form: \(\frac{-6}{4p-5}\).

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Key Concepts

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

Addition of Rational Expressions
When adding rational expressions, it's important to first ensure that the denominators are the same. This involves rewriting the expression so both terms have a common denominator.
In this exercise, we started with \(\frac{3}{4p-5} + \frac{9}{5-4p}\). Notice that \((4p - 5)\) and \((5 - 4p)\) are very similar; they are opposites of each other.
We rewrite \(\frac{9}{5-4p}\) as \(\frac{9}{-(4p-5)}\), which simplifies to \(-\frac{9}{4p-5}\). This way, both fractions now share the same denominator \((4p - 5)\).
Simplifying Rational Expressions
Simplifying rational expressions is crucial for making them easier to add or subtract. This typically involves factoring denominators and numerators where possible. In our case, once the denominators are matched, we can add the numerators directly.
After rewriting the fractions, the expression \(\frac{3}{4p-5} + \frac{9}{5-4p}\) becomes \(\frac{3}{4p-5} + \bigg( -\frac{9}{4p-5} \bigg)\).
Combining like terms in the numerator yields \(\frac{3 - 9}{4p-5}\), which simplifies to \(\frac{-6}{4p-5} \). In this instance, further simplification isn't necessary as \(-6\) and \(4p-5\) do not share common factors.
Common Denominators
Finding a common denominator is one of the most critical steps in the addition or subtraction of rational expressions.
In our example, recognizing that \(4p-5\) and \(5-4p\) are opposites helped us quickly find a common denominator.
After rewriting \(\frac{9}{5-4p}\) to \(-\frac{9}{4p-5}\), the fractions could be added or subtracted easily.
The key takeaway is to always look for equivalent forms of denominators and rewrite them accordingly to simplify your calculations.

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Most popular questions from this chapter

Solve each equation for \(k\) $$ 200=15 k $$

Solve each problem involving direct or inverse variation. If \(d\) varies directly as \(t,\) and \(d=150\) when \(t=3,\) find \(d\) when \(t=5\)

Solve each problem involving direct or inverse variation. If \(d\) varies directly as \(r,\) and \(d=200\) when \(r=40,\) find \(d\) when \(r=60\)

Multiply or divide as indicated. Write each answer in lowest terms. See Section 1.1 $$ \frac{7}{12} \div \frac{15}{4} $$

Determine whether each equation represents direct or inverse variation. $$ y=\frac{12}{x^{2}} $$
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