Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. 1/(x + 1) > 2/(x - 1)

Verified step by step guidance

1. Start by finding a common denominator for the fractions on both sides of the inequality. The common denominator for $\frac{1}{x+1}$ and $\frac{2}{x-1}$ is $(x+1)(x-1)$.
2. Rewrite each fraction with the common denominator: $\frac{1}{x+1}=\frac{x-1}{(x+1)(x-1)}$ and $\frac{2}{x-1}=\frac{2(x+1)}{(x+1)(x-1)}$.
3. Set up the inequality with the common denominators: $\frac{x-1}{(x+1)(x-1)}>\frac{2(x+1)}{(x+1)(x-1)}$.
4. Since the denominators are the same, focus on the numerators: $x-1>2(x+1)$.
5. Solve the inequality $x-1>2x+2$ by isolating $x$ on one side.

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not equal. They can be represented using symbols such as '>', '<', '≥', and '≤'. Solving inequalities involves finding the values of the variable that make the inequality true, which often requires manipulating the expressions similarly to equations but with special attention to the direction of the inequality when multiplying or dividing by negative numbers.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. In the given inequality, the expressions 1/(x + 1) and 2/(x - 1) are rational. Understanding how to manipulate these expressions, including finding common denominators and identifying restrictions (like values that make the denominator zero), is crucial for solving the inequality correctly.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the values that satisfy the inequality. This involves marking open or closed circles to indicate whether endpoints are included (closed) or excluded (open) in the solution. Understanding how to interpret and represent the solution set graphically helps in visualizing the range of values that meet the conditions of the inequality.