Problem 20 - Algebra A Combined Function

Question

Solve. Write the solution set in interval notation. $$ \frac{x+1}{x-4} \leq 0 $$

Step-by-Step Solution

Answer

Solution: \([-1, 4)\).

Step 1: Identify Critical Points

Critical points occur where the numerator of the inequality is zero or where the denominator is zero. For the given inequality \( \frac{x+1}{x-4} \leq 0 \), set the numerator and denominator equal to zero separately.\\(x+1=0\) gives \(x=-1\). \\(x-4=0\) gives \(x=4\).\Thus, the critical points are \(x=-1\) and \(x=4\).

Step 2: Determine Test Intervals

The critical points \(x=-1\) and \(x=4\) divide the number line into intervals: \((-\infty, -1)\), \([-1, 4)\), and \((4, \infty)\).

Step 3: Test Each Interval

Choose a test point from each interval and substitute it into the inequality to determine if it is part of the solution set.\- For \((-\infty, -1)\), choose \(x=-2\): \(\frac{-2+1}{-2-4} = \frac{-1}{-6} = \frac{1}{6} > 0\), not part of the solution.\- For \([-1, 4)\), choose \(x=0\): \(\frac{0+1}{0-4} = \frac{1}{-4} = -\frac{1}{4} \leq 0\), part of the solution. \- For \((4, \infty)\), choose \(x=5\): \(\frac{5+1}{5-4} = 6 > 0\), not part of the solution.

Step 4: Check the Boundary Points

Consider the critical points \(-1\) and \(4\). \For \(x=-1\): \(\frac{-1+1}{-1-4} = 0\), which is included as \(0 \leq 0\).\For \(x=4\), the expression is undefined since it leads to division by zero, so \(x=4\) is not included in the solution set.

Step 5: Write Solution in Interval Notation

Combine the information from steps 3 and 4 to write the solution set, which includes \([-1, 4)\). The interval where the inequality holds true is \([-1, 4)\).

Key Concepts

Interval Notation

Interval notation is a simple way of describing a set of numbers along the number line. It helps in specifying which parts of the number line are included in a particular solution set. In our exercise, interval notation is used to express the solutions for the inequality \( \frac{x+1}{x-4} \leq 0 \).

In interval notation:

• Brackets \([ \text{and} ]\) are used to include endpoints. For instance, \([-1\) indicates that \(-1\) is part of the solution set.
• Parentheses \(( \text{and} )\) are used to exclude endpoints, such as in \((4, \infty)\), meaning 4 is not part of the solution.

For this problem, the solution is written as \([-1, 4)\), meaning \(x\) includes \(-1\) and goes up to, but does not include, 4.

It's a concise way to show that any number from \(-1\) to \(4\) satisfies the given inequality when the conditions of the endpoints are respected.

Critical Points

Critical points are values that can change the direction of inequality expressions. These points occur where either the numerator or the denominator of a rational inequality equals zero. In our given inequality \( \frac{x+1}{x-4} \leq 0 \), the critical points are found by setting \(x+1=0\) and \(x-4=0\).

• The numerator, \(x+1=0\), leads to a critical point at \(x=-1\).
• The denominator, \(x-4=0\), leads to another critical point at \(x=4\).

These critical points \(x=-1\) and \(x=4\) divide the number line into different intervals, which are then further tested to see where the inequality holds true.

Understanding critical points is essential because they indicate where the potential changes in sign or undefined behavior occur, which influences the validity of the inequality in different segments.

Test Intervals

Once we have identified the critical points, we need to divide the number line into test intervals. Test intervals help determine where inequalities are satisfied and allow us to pick distinct sections to analyze the expression. The function \( \frac{x+1}{x-4} \leq 0 \) is affected by tested intervals formed between critical points.

The critical points \(-1\) and \(4\) split the number line into:

• \((-\infty, -1)\)
• \([-1, 4)\)
• \((4, \infty)\)

In each of these intervals, a representative test value is chosen to determine if the inequality holds.

For instance, choosing \(x=-2\) for the interval \((-\infty, -1)\) reveals that the inequality does not hold here. Analyzing the behavior within these intervals is crucial for confirming which values satisfy the original inequality.

Boundary Points

Boundary points are the specific ends of intervals, often derived from critical points. For inequalities, understanding the behaviour of these points is important, especially for determining inclusion in solution sets.

In the problem, we have the boundary points \(-1\) and \(4\). Here's how each boundary affects the solution:

• At \(x=-1\), substituting into the inequality yields zero: \( \frac{-1+1}{-1-4} = 0 \). Since the inequality is \( \leq 0\), \(-1\) is included in the solution.
• At \(x=4\), the denominator becomes zero, making the expression undefined. Therefore, \(x=4\) is not included in the solution set.

These evaluations help refine the final interval notation, indicating precisely which numbers meet the conditions of the inequality. Recognizing boundary points ensures utmost clarity in delineating solution sets for inequalities.