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Chapter 8: Problem 34

Rational Inequalities Solve. $$\frac{1}{x+4}>0$$

Short Answer

Expert verified
The solution is \(x > -4\).

Step by step solution

01

Identify the domain of the inequality

The denominator of the rational inequality, in this case, is \(x+4\). For the inequality \(\frac{1}{x+4}>0\), the expression \(x+4\) should not be equal to zero. Thus, \(x eq -4\). So, the domain of the inequality is all real numbers except \-4\.
02

Determine when the rational expression is positive

The rational expression \(\frac{1}{x+4}\) will be positive when the denominator \(x+4\) is positive. Since the numerator is always positive (as it is 1), we need \(x+4 > 0\) to make the entire fraction positive.
03

Solve for \(x+4 > 0\)

To solve \(x+4 > 0\), subtract 4 from both sides: \(x > -4\).
04

Combine the solution with the domain restrictions

Combining \(x > -4\) with the domain restriction \(x eq -4\), the solution set is unchanged: \(x > -4\).

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Key Concepts

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

solving inequalities
Solving inequalities involves finding the set of values that makes the inequality true. Here, instead of seeking a number where both sides balance out (like in an equation), we need to determine a range of values. Consider an inequality like \(\frac{1}{x+4}>0\). To solve this, follow these steps:
  • Identify the critical points where the expression becomes undefined or zero.
  • Determine the intervals created by these points on a number line.
  • Test values from each interval to see where the inequality holds true.
In our example, the interval testing reveals that \(\frac{1}{x+4}\) is positive when \(x>-4\).
inequality domain
The domain of an inequality refers to all possible values for the variable that don't violate any conditions, such as making a denominator zero. For \( \frac{1}{x+4} > 0 \), we identify where the denominator \(x+4\) is zero to avoid undefined expressions.
  • Set the denominator equal to zero: \(x+4=0\).
  • Solve for the variable to find exclusion points: \(x e -4\).
The domain is then all real numbers except the excluded point, hence \(x e -4\).
positive rational expressions
Positive rational expressions are fractions that result in positive values. For the expression \( \frac{1}{x+4} \) to be positive:
  • The numerator (top part) must be positive. In this case, 1 is always positive.
  • The denominator (bottom part) must also be positive, since two positive numbers result in a positive fraction.
Thus, \( x+4> 0 \) leads us to find where \( x > -4\). By analyzing combined conditions, we solve for positive rational expressions accurately.

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Most popular questions from this chapter

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