In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5

Verified step by step guidance

1. Rewrite the inequality in standard form by subtracting 5 from both sides: \( \frac{x + 3}{x - 4} - 5 \leq 0 \). Combine the terms into a single fraction by finding a common denominator: \( \frac{x + 3 - 5(x - 4)}{x - 4} \leq 0 \).
2. Simplify the numerator of the fraction: \( x + 3 - 5(x - 4) = x + 3 - 5x + 20 = -4x + 23 \). The inequality becomes \( \frac{-4x + 23}{x - 4} \leq 0 \).
3. Determine the critical points by setting the numerator and denominator equal to zero: \( -4x + 23 = 0 \) gives \( x = \frac{23}{4} \), and \( x - 4 = 0 \) gives \( x = 4 \). These critical points divide the number line into intervals.
4. Test the sign of the fraction in each interval created by the critical points (e.g., \( (-\infty, 4) \), \( (4, \frac{23}{4}) \), and \( (\frac{23}{4}, \infty) \)) by substituting test values into the fraction \( \frac{-4x + 23}{x - 4} \). Determine where the fraction is less than or equal to zero.
5. Include the critical points in the solution set if they satisfy the inequality. For \( x = \frac{23}{4} \), the numerator is zero, so it satisfies the inequality. For \( x = 4 \), the denominator is zero, so it is excluded. Represent the solution set on a real number line, shading the appropriate intervals and marking any included points.

Key Concepts

Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like ≤, ≥, <, or >. In this case, the inequality (x + 3)/(x - 4) ≤ 5 indicates that the fraction on the left must be less than or equal to 5. Understanding how to manipulate and solve inequalities is crucial for finding the solution set.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In the given inequality, (x + 3)/(x - 4) is a rational expression. To solve the inequality, one must consider the behavior of the expression, including its domain and any restrictions, such as values that make the denominator zero.

Graphing Solution Sets

Graphing the solution set on a real number line visually represents the values of x that satisfy the inequality. This involves identifying critical points, such as where the expression equals 5 or is undefined, and determining intervals where the inequality holds true. Properly shading the solution set helps in understanding the range of valid solutions.