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Chapter 7: Problem 16

List all numbers for which each rational expression is undefined. $$ \frac{a-8}{a+7} $$

Short Answer

Expert verified
The expression is undefined when \(a = -7\).

Step by step solution

01

Identify the Denominator

Locate the denominator of the rational expression. In this case, the denominator is \(a + 7\).
02

Set the Denominator to Zero

To determine when the expression is undefined, set the denominator equal to zero: \(a + 7 = 0\).
03

Solve for 'a'

Solve the equation \(a + 7 = 0\) to find the value of 'a'. Subtract 7 from both sides to isolate 'a': \(a = -7\).
04

List the Solution

The rational expression \(\frac{a-8}{a+7}\) is undefined when \(a = -7\).

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Key Concepts

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

rational expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. These expressions represent ratios of polynomials and are a core concept in algebra. Understanding rational expressions involves managing both the numerator and the denominator efficiently.

Here are key points about rational expressions:
  • A rational expression can look like \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
  • For a rational expression to be defined, the denominator \(Q(x)\) must not equal zero.
  • When \(Q(x)\) equals zero, the expression becomes undefined.

These expressions are widely used in various mathematical processes, including solving equations and simplifying complex fractions. The key takeaway is the importance of the denominator in ensuring the expression remains valid.
denominator
The denominator of a fraction is the bottom part, which indicates how many equal parts the whole is divided into. In a rational expression, the denominator is critical in determining whether the expression is defined or undefined.

Trying to grasp the role of the denominator can be simplified through a few essential points:
  • Identify the denominator by determining the polynomial or number at the bottom of the fraction.
  • Ensure the denominator is never equal to zero, as this results in an undefined expression.
  • Solve for the variable in the denominator to find the values that make the expression undefined, as seen in the example above where \(a + 7 = 0\).

For instance, if we look at the expression \(\frac{a-8}{a+7}\), we see that the denominator is \(a + 7\). Setting this equal to zero and solving for \(a\) helps us conclude that the expression is undefined when \(a = -7\).
solving equations
Solving equations is a fundamental process in algebra that involves finding the value of variables that make the equation true. When dealing with rational expressions, solving equations often includes ensuring that the denominator does not equal zero.

Let's understand how equations are solved in a few steps:
  • Identify the equation to solve. In this example, the equation was \(\frac{a-8}{a+7}\).
  • Set the denominator equal to zero to find values that may make the expression undefined.
  • Solve the resulting equation. For \(a + 7 = 0\), we subtract 7 from both sides, leading to \(a = -7\).

These steps are not just exclusive to finding undefined values but are general methods for solving various algebraic equations. Always check the solutions to ensure they make the original equation valid.

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Most popular questions from this chapter

A device used in golf to estimate the distance \(d\) to a hole measures the size \(s\) that the 7 -ft pin appears to be in a viewfinder. The viewfinder uses the principle, diagrammed here, that \(s\) gets bigger when \(d\) gets smaller. If \(s=0.56\) in. when \(d=50\) yd, find an equation of variation that expresses \(d\) as a function of \(s .\) What is \(d\) when \(s=0.40 \mathrm{in} . ?\) PICTURE NOT AVAILABLE

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Escalators. Together, a \(100-\mathrm{cm}\) wide escalator and a \(60-\mathrm{cm}\) wide escalator can empty a 1575 -person auditorium in 14 min. The wider escalator moves twice as many people as the narrower one. How many people per hour does the \(60-\mathrm{cm}\) wide escalator move?

Find the LCM. $$4(x-1), 8(x-1)$$
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