Problem 56 - Intermediate Algebra

Question

Solve each inequality. $$ \frac{x+3}{x-4} \geq 1 $$

Step-by-Step Solution

Answer

The solution is \( x > 4 \).

Step 1: Set Up the Inequality

Start by setting up the inequality as given: \( \frac{x+3}{x-4} \geq 1 \). To proceed, write the inequality in a form that compares it to zero: \( \frac{x+3}{x-4} - 1 \geq 0 \).

Step 2: Simplify the Expression

Rewrite the inequality as a single fraction: \( \frac{x+3}{x-4} - \frac{x-4}{x-4} \geq 0 \). Combine the fractions: \( \frac{x+3-(x-4)}{x-4} \geq 0 \). This simplifies to \( \frac{7}{x-4} \geq 0 \).

Step 3: Determine Critical Points

Identify the points where the inequality changes. Since the expression is a fraction, consider where the numerator is zero. However, 7 is always non-zero. Consider the denominator instead: \( x-4 = 0 \) gives \( x = 4 \). This point makes the denominator zero and isn't included in the solution since it makes the expression undefined.

Step 4: Analyze the Sign of the Expression

Check intervals around the critical point to determine where the inequality holds true. Choose a test point from each interval: \( x = 3 \) for \( x < 4 \) and \( x = 5 \) for \( x > 4 \). For \( x = 3 \), \( \frac{7}{3-4} = -7 \) (negative). For \( x = 5 \), \( \frac{7}{5-4} = 7 \) (positive).

Step 5: Write the Solution

Since we need \( \frac{7}{x-4} \geq 0 \) and it is only positive for \( x > 4 \), the final solution is \( x > 4 \).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions containing fractions where one expression is compared to another using inequality symbols like ">", "<", "≥", or "≤". These fractions often have variables in the numerator, denominator, or both. Solving such inequalities typically requires manipulation to form a zero-inequality. This involves moving all terms to one side, usually resulting in a single rational expression that can be analyzed further. When dealing with inequalities like \( \frac{x+3}{x-4} \geq 1 \), we transform it to \( \frac{x+3-(x-4)}{x-4} \geq 0 \) for manageable analysis.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In our original exercise, \( \frac{x+3}{x-4} \) represents such an expression. Simplifying this expression involves understanding how to combine and manipulate these components effectively. The simplification process transforms the original inequality to \( \frac{7}{x-4} \ geq 0 \), which is easier to analyze for solving the inequality.

Critical Points

Critical points are the values of the variable that make the denominator zero or significantly affect the direction of the inequality. They are important because these points are where the rational expression could change from positive to negative or vice versa. In this context, considering \( x-4 \), the critical point is found by setting the denominator equal to zero, yielding \( x=4 \). This value makes the original inequality expression undefined, meaning it is not part of the solution.

Interval Testing

Interval testing is a strategy to determine where a rational inequality holds true. It involves selecting test points in each interval created by the critical points and substituting them into the simplified inequality to check the sign. For our inequality, the critical point is \( x=4 \), creating intervals \((-\infty, 4)\) and \((4, \infty)\). Choosing test points like \( x=3 \) for \((x < 4)\) and \( x=5 \) for \((x > 4)\) helped analyze the sign of \( \frac{7}{x-4} \), showing it is negative in the first interval and positive in the second, thus determining where the original inequality holds.