Problem 75 - Precalculus Student Solutions Manual 5th

Question

Solve each rational inequality. Write each solution set in interval notation. $$\frac{-4}{1-x}<5$$

Step-by-Step Solution

Answer

No solution.

Step 1: Isolate the Rational Expression

Start with the inequality \ \( \frac{-4}{1-x} < 5 \). We want to isolate the rational expression on one side of the inequality. The inequality is already set up this way, so proceed to the next step.

Step 2: Analyze the Rational Expression

Denote the rational expression as \( \frac{-4}{1 - x} \). Consider the two cases separately: \(1 - x > 0\) and \(1 - x < 0\), as the inequality changes direction depending on the sign of the denominator.

Step 3: Case 1 - Denominator Positive

For \(1 - x > 0\), solve the inequality \ \( \frac{-4}{1 - x} < 5 \). Since the denominator (\(1 - x\)) is positive, the inequality remains the same. Multiply both sides by \(1 - x\) to get \ \( -4 < 5(1 - x) \).

Step 4: Solve the Inequality with Positive Denominator

Expand and solve for \(x\): \ -4 < 5 - 5x \ Rearrange terms: \ -4 - 5 < -5x \ -9 < -5x \ Divide by -5 (flipping the inequality): \ \frac{9}{5} > x \ or \ x < \frac{9}{5}.

Step 5: Case 2 - Denominator Negative

For \(1 - x < 0\), solve the inequality \ \( \frac{-4}{1 - x} < 5 \). Since the denominator (\(1 - x\)) is negative, multiply both sides by \(1 - x\) and reverse the inequality direction: \ \frac{-4}{1 - x} > 5.

Step 6: Solve the Inequality with Negative Denominator

Multiply both sides by \(1 - x\): \ -4 > 5(1 - x) \ Expand and solve for \(x\): \ -4 > 5 - 5x \ Rearrange terms: \ -4 - 5 > -5x \ -9 > -5x \ Divide by -5 (flipping the inequality): \ \frac{9}{5} < x \ or \ x > \frac{9}{5}.

Step 7: Combine Solution Sets

Combine the solutions from both cases. For \(1 - x > 0\), we have \( x < \frac{9}{5} \). For \(1 - x < 0\), we have \( x > \frac{9}{5} \). Since there is no overlap and \( x \) cannot be equal to \( \frac{9}{5} \) (as the denominator would be zero), there is no solution.

Key Concepts

Interval Notation

When you solve inequalities, expressing the solution set is a crucial step. Interval notation is used to express the range of values that satisfy the inequality in a concise format. For instance:

• '(' or ')' denote that the endpoint is not included.
• '[' or ']' denote that the endpoint is included.

In our example, solving the rational inequality involved identifying two cases but found that there were no overlapping values. This means there is no range of x values that satisfy the inequality. Thus, the solution set is the empty set, often denoted by \(\emptyset\).

Rational Expressions

A rational expression is basically a fraction where both the numerator and the denominator are polynomials. In our example, we have the expression \(\frac{-4}{1 - x}\). Understanding rational expressions is important because they can shift dramatically depending on the value of the variable, particularly as the denominator approaches zero. You must:

• Consider the values that make the denominator zero, to avoid undefined expressions.
• Analyze the behavior of the expression for ranges where the denominator is positive or negative separately.

Here, the critical value is \(x = 1\), which makes the denominator zero, and it's essential in defining the solution constraints.

Inequalities

Inequalities involve comparisons such as less than, greater than, less than or equal to, and greater than or equal to. Solving rational inequalities requires extra care:

• Identify when to reverse the inequality sign. Multiplying or dividing by a negative number reverses it.
• Analyze the sign of the denominator to determine how to handle the inequality.

In our problem, we divided the solution into two cases by analyzing when \(1 - x\) was positive or negative. By splitting it into these cases, we correctly handled the direction of the inequality, but the two cases did not overlap, leading to no solution.