Problem 26 - Algebra A Combined Function

Question

Solve. Write the solution set in interval notation. $$ \frac{x-1}{x+4}>2 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -9)\) in interval notation.

Step 1: Set Up the Inequality

We are given the inequality \( \frac{x-1}{x+4} > 2 \). To eliminate the fraction, we multiply both sides by \( x+4 \), which is assumed to be non-zero. This gives us the inequality: \( x - 1 > 2(x + 4) \).

Step 2: Simplify the Right Side

Expand the right side expression: \( 2(x + 4) = 2x + 8 \). The inequality now becomes \( x - 1 > 2x + 8 \).

Step 3: Solve for x

Subtract \( x \) from both sides of the inequality: \( -1 > x + 8 \). Then, subtract 8 from both sides to isolate \( x \): \( -1 - 8 > x \), which simplifies to \( -9 > x \), or equivalently, \( x < -9 \).

Step 4: Consider Undefined Points

Consider when the denominator of the initial fraction, \( x+4 \), is zero, i.e., \( x = -4 \). The function is undefined at this point, so \( x = -4 \) should not be included in the solution set. Thus, the solution is not continuous over its entire domain.

Step 5: Write Solution in Interval Notation

Since \( x < -9 \) and we must exclude \( x = -4 \), the solution set is written in interval notation as \( (-\infty, -9) \), noting that \( x = -4 \) falls outside this interval and does not affect the solution set.

Key Concepts

Understanding Interval Notation

Interval notation is a way of expressing ranges of values that solutions can take. It's often used in solving inequalities, as it helps to clearly show all possible solutions in a compact format.
For example, the interval

• \((a, b)\) includes all numbers greater than \(a\) and less than \(b\), excluding \(a\) and \(b\) themselves. Called an open interval.
• \([a, b]\) includes all numbers from \(a\) to \(b\), including both \(a\) and \(b\). This is a closed interval because it contains both endpoints.
• \((a, b]\) and \([a, b)\) are half-open intervals, including only one of the endpoints.

In our solution, the notation \((-\infty, -9)\) means all real numbers less than \(-9\). \(-\infty\) indicates there is no lower bound, and the open parenthesis shows \(-9\) is not included.
This notation is particularly useful as it succinctly communicates the range of possible solutions without listing each number.
Understanding how to read and write interval notation is crucial when solving inequalities.

What are Rational Inequalities?

Rational inequalities involve expressions containing fractions with variables in the numerator, denominator, or both. They are called rational because they involve ratios of polynomial expressions.
Unlike standard linear inequalities, rational inequalities often have points where the expression is undefined, such as division by zero. These can affect the solution and must be considered.

• Inequalities such as \(\frac{x}{y} > c\) require extra care in consideration of potential restrictions on \(x\) and \(y\).
• The inequality \(\frac{x-1}{x+4} > 2\) in our example is rational because it contains a fraction with a variable denominator.

When solving rational inequalities, be sure to check where the denominator becomes zero. This is critical because those points need to be excluded from the solution set, as seen in the example where \(x=-4\) was not included.

Exploring Solution Sets

Solution sets are the sets of values that satisfy an equation or inequality. In more straightforward terms, they are the answers to a problem.
When dealing with inequalities, solution sets can be infinite, consisting of all numbers within a certain range.
In our problem, the solution set is expressed as \((-\infty, -9)\). This tells us:

• All numbers less than \(-9\) will make the original inequality true.
• Since \(x\) cannot be \(-4\), \(-9 < x < -4\) isn't a concern in this context because the interval \((-\infty, -9)\) eliminates these values.

The solution set is essential because it effectively communicates any restrictions needed to satisfy the conditions of the inequality.

Essential Inequality Solving Techniques

Solving inequalities involves a set of techniques that ensure the solution is correct and accurate. Here are some fundamental principles:

• Eliminate fractions: Multiply both sides by the denominator to get rid of any fractions, taking care not to multiply by zero.
• Isolate the variable: Use basic algebra to rearrange terms, bringing the variable of interest alone on one side of the inequality.
• Consider critical points: With rational inequalities, identify points where the expression is zero or undefined.
• Test intervals: After finding critical points, test regions around these points to determine where the inequality holds true.

In the initial exercise, multiplying both sides by \(x+4\) and then simplifying helped eliminate the fraction. However, defining where the expression is undefined through identifying \(x=-4\) was crucial. Finally, writing the solution using interval notation clarified which region of \(x\) made the inequality valid.