Solve each inequality in Exercises 86–91 using a graphing utility. 1/(x + 1) ≤ 2/(x + 4)

Verified step by step guidance

1. Start by rewriting the inequality to have all terms on one side: \(\frac{1}{x + 1} - \frac{2}{x + 4} \leq 0\).
2. Find a common denominator to combine the fractions: the common denominator is \((x + 1)(x + 4)\), so rewrite the expression as \(\frac{(x + 4) - 2(x + 1)}{(x + 1)(x + 4)} \leq 0\).
3. Simplify the numerator: expand and combine like terms to get \(\frac{x + 4 - 2x - 2}{(x + 1)(x + 4)} \leq 0\), which simplifies to \(\frac{-x + 2}{(x + 1)(x + 4)} \leq 0\).
4. Identify the critical points by setting the numerator and denominator equal to zero: numerator zero at \(x = 2\), denominator zero at \(x = -1\) and \(x = -4\). These points divide the number line into intervals to test.
5. Use a graphing utility to plot the function \(f(x) = \frac{-x + 2}{(x + 1)(x + 4)}\) and determine where \(f(x) \leq 0\) by observing the graph and considering the domain restrictions where the denominator is not zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values of the variable that make the inequality true, while considering restrictions where the denominator is zero to avoid undefined expressions.

Domain Restrictions

Domain restrictions are values that make the denominator zero, causing the expression to be undefined. Identifying these values is crucial before solving inequalities, as they split the number line into intervals for testing solutions.

Graphing Utility for Inequalities

A graphing utility helps visualize the functions involved in the inequality by plotting their graphs. It allows for identifying where one function is less than or equal to another, making it easier to determine solution intervals accurately.