Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{x+4}{x-3} \leq 0 $$

Step-by-Step Solution

Answer

The solution set is \([-4, 3)\).

Step 1: Identify Critical Points

The critical points of the inequality come from the values that make the numerator zero and the denominator zero. First, find when the numerator is zero by solving \( x + 4 = 0 \), giving \( x = -4 \). Next, find when the denominator is zero by solving \( x - 3 = 0 \), giving \( x = 3 \). These critical points are \( x = -4 \) and \( x = 3 \).

Step 2: Test Intervals Around Critical Points

Determine the intervals created by the critical points: \( (-\infty, -4) \), \( (-4, 3) \), and \( (3, \infty) \). Choose a test value from each interval and insert it into the inequality to check the sign of \( \frac{x+4}{x-3} \). Use \( x = -5 \) for the first interval; it results in positive, use \( x = 0 \) for the second interval; it results in negative, and use \( x = 4 \) for the third interval; it results in positive.

Step 3: Determine Inequality Validity

Since we need the result to be \( \leq 0 \), take intervals where the inequality is zero or negative. The numerator is zero at \( x = -4 \), which satisfies the inequality. The interval \( (-4, 3) \) results in a negative value thus also satisfies the inequality, but \( x = 3 \) itself is excluded because it makes the denominator zero.

Step 4: Write Solution in Interval Notation

The solution combines the numbers and sets from valid intervals: \( [-4, 3) \). Here, \(-4\) is included because it satisfies the \( = 0 \) part of the inequality, and \(3\) is excluded as dividing by zero is undefined.

Step 5: Sketch the Graph

Draw a number line. Mark \( x = -4 \) with a closed circle, indicating it is included in the solution. Mark \( x = 3 \) with an open circle to show it is not included. Shade the region between \( x = -4 \) and \( x = 3 \) to represent \( -4 \leq x < 3 \).

Key Concepts

Interval Notation

Interval notation is a simple way to represent the set of numbers that satisfy an inequality. It communicates which numbers are part of the solution and which are not through brackets.

• Curved brackets \(( , )\) indicate that a number is not included in the interval. This is often used when the inequality is strict, such as \(x < a\) or \(x > b\).
• Square brackets \([ , ]\) show that a number is included in the interval. It is used when the number satisfies the inequality, such as in \(x \leq a\) or \(x \geq b\).

In the given inequality, \( \frac{x+4}{x-3} \leq 0 \), we find the critical points and test intervals to determine the solution. After testing the intervals, the solution is given as \([-4, 3)\). Here, \(-4\) is included because it gives the expression zero, matching the inequality condition \( \leq 0 \). On the other hand, \(3\) is not included, as it would make the denominator zero, which is undefined. This interval notation precisely shows the continuous solution set from \(-4\) up to, but not including, \(3\).

Critical Points

Critical points in inequalities, particularly when dealing with rational expressions, are points that can potentially change the sign of the expression. They are found by determining where the expression is zero or undefined.

• To find critical points, solve for when the numerator is zero. For the inequality \(\frac{x+4}{x-3} \leq 0\), solve \(x+4=0\) to get \(x=-4\).
• Also solve for when the denominator is zero, yielding \(x-3=0\) or \(x=3\).

These two values split the number line into distinct intervals. The critical points \(x=-4\) and \(x=3\) are crucial because they can indicate where the expression transitions from positive to negative or vice-versa. Understanding critical points helps in selecting the correct intervals that satisfy a given inequality.

Number Line Graphing

Number line graphing visualizes the solution to an inequality. It effectively shows which parts of the number line are part of the solution set.

• Start by drawing a horizontal line and marking the critical points found earlier, \(-4\) and \(3\).
• Use a closed circle at \(-4\) to show it is included in the solution, as the original inequality allows for equality, i.e., \( \leq 0 \).
• Place an open circle at \(3\) to indicate it is not part of the solution set, as the expression is undefined here.
• Shade the area between \(-4\) and \(3\), demonstrating that all numbers in this interval satisfy the inequality \(-4 \leq x < 3\).

Graphing on a number line provides a clear, visual representation of the numbers that make the inequality true. It's a helpful tool to cross-check your interval notation solution.