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. Step 1: Start by rewriting the inequality $\frac{x+3}{x-4}\le 5$ in a form that is easier to solve. Subtract 5 from both sides to get $\frac{x+3}{x-4}-5\le 0$.
2. Step 2: Combine the terms on the left side over a common denominator: $\frac{x+3-5(x-4)}{x-4}\le 0$. Simplify the numerator to get $\frac{x+3-5x+20}{x-4}\le 0$.
3. Step 3: Simplify the expression further: $\frac{-4x+23}{x-4}\le 0$.
4. Step 4: Identify the critical points by setting the numerator and denominator equal to zero: $-4x+23=0$ and $x-4=0$. Solve these equations to find the critical points.
5. Step 5: Use the critical points to divide the number line into intervals. Test each interval to determine where the inequality holds true. Remember to consider the behavior at the critical points and whether they are included in the solution set.

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 number line helps convey the complete solution set.