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Chapter 8: Problem 37

Rational Inequalities Solve. $$\frac{x+1}{x+6} \geq 1$$

Short Answer

Expert verified
The solution is \( x \leq -6 \).

Step by step solution

01

- Understand the Inequality

The inequality to solve is: \( \frac{x+1}{x+6} \geq 1\). This means we need to find all values of \( x \) for which this inequality holds true.
02

- Rewrite the Inequality

Subtract 1 from both sides of the inequality to combine the fractions: \( \frac{x+1}{x+6} - 1 \geq 0\). Rewrite the left side with a common denominator: \( \frac{x+1 - (x+6)}{x+6} \geq 0\).
03

- Simplify the Inequality

Simplify the expression in the numerator: \( \frac{x + 1 - x - 6}{x+6} \geq 0\). Which simplifies to: \( \frac{-5}{x+6} \geq 0\).
04

- Recognize the Solution Regions

Analyze the simplified inequality: \( \frac{-5}{x+6} \geq 0\). Since \( -5 \) is always negative, and the fraction can only be positive when the denominator (\( x+6 \)) is negative or zero, we must find the interval where \( x+6 \leq 0 \).
05

- Solve for x

Solve the inequality: \( x + 6 \leq 0\). Subtract 6 from both sides: \( x \leq -6\).
06

- Final Solution in Interval Notation

Finally, represent the solution set in interval notation: \( (-\infty, -6] \).

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Key Concepts

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

inequality
An inequality is a mathematical statement that compares two expressions using inequality signs like >, <, 鈮�, and 鈮�. For instance, in the problem \( \frac{x+1}{x+6} \geq 1 \), we are dealing with the 'greater than or equal to' sign (鈮�). Inequalities show that one side is either larger or equal to the other side. This helps us understand the range of values that the variable x can take to make the inequality true.
rational expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, \( \frac{x+1}{x+6} \) is a rational expression because both \ x+1 \ and \ x+6 \ are polynomials. Solving inequalities involving rational expressions requires combining and simplifying these expressions to figure out where they meet the inequality conditions.
interval notation
Interval notation is a shorthand used to describe sets of numbers lying between two endpoints. For the solution \( (-\infty, -6] \), we use interval notation to explain that x can be any value from negative infinity up to and including -6. Square brackets [ ] mean the end number is included, while parentheses ( ) mean it's not included. So \( (-6, -\infty) \) would mean x is more than -6 but not including -6.
solving inequalities
To solve inequalities, follow these steps:

  • Rewrite the inequality if necessary to make it easier to solve.
  • Simplify expressions on both sides.
  • Isolate the variable to one side of the inequality.
For the given problem, we simplified \( \frac{x+1}{x+6} \geq 1 \) by subtracting 1 from both sides and then combining the fractions. After simplifying and analyzing, we concluded that \( x+6 \leq 0\) gives the solution \( x \leq -6 \). Convert this into interval notation to get the final answer: \( (-\infty, -6] \).

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