Problem 68 - Algebra and Trigonometry

Question

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{1}{x+1}>\frac{2}{x-1} $$

Step-by-Step Solution

Answer

The solution set of the inequality \(\frac{1}{x+1}>\frac{2}{x-1}\) is \(( -∞, -3)\)

Step 1: Clear the Fractions

Multiply each side of the inequality by \((x+1)(x-1)\) to clear the fractions. \((x+1)(x-1)\times \frac{1}{x+1} > \((x+1)(x-1)\times \frac{2}{x-1}, resulting in \((x - 1) > 2(x + 1)\).

Step 2: Simplify the Inequality

Distribute the 2 on the right hand side to get \(x - 1 > 2x + 2.\) Move all terms to one side to form a quadratic inequality: \(2x + 3 < x\).

Step 3: Solve the Quadratic Inequality

Subtract x from both sides to isolate the variable on one side: \( x + 3 < 0.\) Solution to the inequality is \( x < -3\).

Step 4: Test Intervals around the Critical Point

Choose test points from each of the intervals divided by the critical point -3. For x < -3, choose, for instance, x = -4. Substituting x = -4 into the original inequality gives a true statement. Thus, the interval \(( -∞, -3)\) satisfies the inequality.

Step 5: Graph the Solution Set

On the number line, an open circle (indicating 'less than') is plotted at -3 and an arrow points towards -∞. This represents the solution set for the inequality.