Problem 46 - College Algebra

Question

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{1}{x+1}>\frac{2}{x-1} $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, -3)\).

Step 1: Find common denominator

Common denominator of \(x+1\) and \(x-1\) is \((x+1)(x-1)\), which simplifies to \(x^2-1\). Multiply each fraction by \(x^2-1\) to remove the denominators. This gives: \((x-1) > 2(x+1)\)

Step 2: Simplify and solve the inequality

Expanding the brackets we get \(x-1 > 2x+2\). Simplifying this further gives: \(-x > 3\). To keep the inequality sign the same, we multiply each side by -1 and switch the inequality sign to '<'. This gives: \(x < -3\)

Step 3: Find the restriction

To find the restriction in x, we set denominators equal to zero. This gives us \(x=-1, x=1\). Since the given inequality excluded these points, our solution for the inequality \(x < -3\) holds.

Step 4: Interval notation and number line representation

Express the solution as an interval. Our solution \(x < -3\) can be written in interval notation as \((-\infty, -3)\). The graph on the number line will be a ray pointing to the left from the point -3, excluding the point itself.

Key Concepts

Common Denominator

In rational inequalities, finding a common denominator is a crucial step. It allows us to compare fractions without denominators. Given an inequality such as \(\frac{1}{x+1} > \frac{2}{x-1}\), we first find a common denominator for the two fractions. For the terms \(x+1\) and \(x-1\), a common denominator is their product, which is \((x+1)(x-1)\). Simplifying the product, we get \(x^2 - 1\). This common denominator turns our fractions into expressions that can be compared directly.
\[\frac{(x-1)}{x^2 - 1} > \frac{2(x+1)}{x^2 - 1}\]By multiplying both sides by \(x^2 - 1\), we clear the denominators, simplifying the inequality to an easier form: \(x-1 > 2(x+1)\).
This process transforms the problem from a complex inequality involving fractions into a simpler, more straightforward algebraic inequality.

Inequality Solution

After finding a common denominator and clearing it, the next step is solving the resulting inequality. From the transformation made in the previous step, we have the inequality \(x-1 > 2(x+1)\).
Expanding and simplifying this, we get: \[x - 1 > 2x + 2\]
Rearrange terms to isolate the variable \(x\):

• Subtract \(2x\) from both sides: \(-x - 1 > 2\)
• Add \(1\) to both sides: \(-x > 3\)

To solve for \(x\), multiply both sides by \(-1\). Since multiplying by a negative number flips the inequality sign, we have: \[x < -3\] Thus, our inequality solution is \(x < -3\). It’s also important to consider restrictions from the original denominators; here, \(x\) cannot be \(-1\) or \(1\) because these make the original expressions undefined.

Interval Notation

Interval notation is a concise way to describe sets of solutions. It harnesses brackets and parenthesis to denote the start and end of solution intervals on the number line. In our current example, the solution to the inequality is \(x < -3\). In interval notation, this is written as \((-\infty, -3)\).

• Round or open parenthesis, \((\), indicates that the endpoint is not included in the solution set.
• An interval starting with negative infinity, \(\ (-\infty\)\, indicates that the range extends infinitely in the negative direction.
• The number at the end, in this case \(-3\), represents the boundary point that is not part of the solution set.

This notation provides a clear, visual way to interpret which values \(x\) can take, avoiding verbose descriptions.

Number Line Representation

A number line representation visually displays the solutions of inequalities, making it easier to understand at a glance. For the solution \(x < -3\), draw a number line and locate the point \(-3\). Indicate an open circle at \(-3\) to show that \(-3\) itself is not included in the solution.

• Draw a ray or an arrow starting from this point and extending to the left, depicting all numbers less than \(-3\).
• The open circle clearly shows exclusion of the boundary point.
• The arrow signifies the continuation in the negative direction.

This visual aid helps in quickly identifying the solution set and any critical points, aligning with the interval notation \((-\infty, -3)\). It's an essential skill for interpreting and communicating mathematical solutions effectively.