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Chapter 7: Problem 32

Write each rational expression in lowest terms. $$ \frac{4 x-9}{4} $$

Short Answer

Expert verified
\frac{4x - 9}{4} is already in lowest terms.

Step by step solution

01

Identify the Rational Expression

Given the rational expression \(\frac{4x - 9}{4}\), it consists of a numerator (which is \(4x - 9\)) and a denominator (which is 4).
02

Simplify the Numerator and Denominator

Check if there is any common factor between the numerator and the denominator. In this case, the numerator \(4x - 9\) does not have any common factors with the denominator 4.
03

Express in Lowest Terms

Since there are no common factors between the numerator and the denominator, the rational expression is already in its lowest terms. Therefore, the simplified form of \(\frac{4x - 9}{4}\) is \(\frac{4x - 9}{4}\).

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Key Concepts

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

headline of the respective core concept
Rational expressions are fractions where the numerator and the denominator are both polynomials. They are similar to regular fractions but with more complex expressions. For example, \(\frac{4x - 9}{4}\) is a rational expression. The numerator is \(4x - 9\) and the denominator is \4\.

To work with rational expressions, you often need to simplify them. This includes factoring polynomials, canceling common terms, and expressing the result in the simplest form. Just like with regular fractions, the goal is to make the expression as simple as possible while maintaining its value.
headline of the respective core concept
In math, expressing something in its lowest terms means simplifying it as much as possible. For rational expressions, this involves finding and canceling common factors from both the numerator and the denominator.

Consider the rational expression \(\frac{4x - 9}{4}\). Step by step, we first look at the numerator \(4x - 9\) and the denominator \4\. If both had common factors, we could divide them out to simplify the expression. However, in this case, the numerator and denominator have no common factors. Thus, the expression \(4x - 9 / 4\) remains in the simplest form, making it already in its lowest terms.
headline of the respective core concept
Simplification is about making mathematical expressions as straightforward as possible. This usually means finding the simplest equivalent form. For rational expressions like \(\frac{4x - 9}{4}\), the steps to simplify include:
  • Identifying any common factors between the numerator and denominator.
  • Canceling out those common factors.
  • Rewriting the expression in its simplest form.
In our example, since there are no common factors to cancel between \(4x - 9\) and \4\, \(4x - 9 / 4\) is already simplified.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{t^{2}-49}{t^{2}+4 t-21} \cdot \frac{t^{2}+8 t+15}{t^{2}-2 t-35} $$

Solve each problem. The percent of deaths caused by smoking is modeled by the rational function defined by $$ p(x)=\frac{x-1}{x} $$ where \(x\) is the number of times a smoker is more likely to die of lung cancer than a nonsmoker is. This is called the incidence rate. (Source: Walker, A., Observation and Inference: An Introduction to the Methods of Epidemiology, Epidemiology 91影视 Inc.) For example, \(x=10\) means that a smoker is 10 times more likely than a nonsmoker to die of lung cancer. (a) Find \(p(x)\) if \(x\) is 10 (b) For what values of \(x\) is \(p(x)=80 \% ?\) (Hint: Change \(80 \%\) to a decimal.) (c) Can the incidence rate equal \(0 ?\) Explain.

Write each rational expression in lowest terms. $$ \frac{7 x-21}{63-21 x} $$

Concept Check Write each formula using the "language" of variation. For example, the formula for the circumference of a circle, \(C=2 \pi r,\) can be written as "The circumference of a circle varies directly as the length of its radius." \(S=4 \pi r^{2},\) where \(S\) is the surface area of a sphere with radius \(r\)

Concept Check Write each formula using the "language" of variation. For example, the formula for the circumference of a circle, \(C=2 \pi r,\) can be written as "The circumference of a circle varies directly as the length of its radius." \(P=4 s,\) where \(P\) is the perimeter of a square with side of length \(s\)
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