Problem 53 - College Algebra

Question

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{(x+4)(x-1)}{x+2} \leq 0 $$

Step-by-Step Solution

Answer

The solution of the inequality \(\frac{(x+4)(x-1)}{x+2}\leq 0\) in interval notation is \([-4, 1]\).

Step 1: Set the numerator and denominator equal to zero

Setting both the numerator and the denominator equal to zero gives the equations \(x+4 = 0\) and \(x-1 = 0\) for the numerator, and \(x+2 = 0\) for the denominator. Solving these equations gives \(x = -4, 1\) and \(x = -2\) respectively. Discard \(x = -2\) because it makes the denominator equal to zero.

Step 2: Place the critical numbers on a number line

The critical numbers are -4 and 1. They divide the real number line into three intervals: \(-\infty, -4\), \(-4, 1\), and \(1, \infty\).

Step 3: Check the sign of the inequality in each interval

Choose a test number from each interval and substitute into the original inequality. For interval \(-\infty, -4\), choose -5; for \(-4, 1\), choose 0; and for \(1, \infty\), choose 2. This yields positive, negative, and positive results respectively.

Step 4: Write the solution in interval notation

Since the inequality is less than or equal to zero and we want values of x that make it negative or zero, the solution is \([-4, 1]\). Include -4 and 1 because the inequality allows for zero, and -4 and 1 make the inequality equal to zero.

Key Concepts

Critical Numbers

Critical numbers are fundamental when solving rational inequalities. They are essential to identifying the points on the number line that can potentially change the sign of the inequality—specifically, where the numerical expression equals zero or is undefined. For a rational expression like \(\frac{(x+4)(x-1)}{x+2}\), critical numbers come from:

• Setting the numerator \((x+4)(x-1)\) equal to zero gives points where the expression itself could be zero, found at \(x = -4\) and \(x = 1\).
• Setting the denominator \(x+2\) equal to zero identifies points where the expression is undefined. Here, it is \(x = -2\).

In the context of the given inequality, critical numbers \(-4\) and \(1\) divide the number line into distinct intervals to test for sign changes. The denominator value \(x = -2\) is not included in the solution as it causes division by zero, which makes the entire expression undefined. This subtraction results in three specific ranges to consider: \((-\infty, -4), (-4, 1), (1, \infty)\). By working with these values, you ascertain where on the number line the inequality holds true.

Interval Notation

Interval notation provides a concise way to express the solution sets of inequalities. It describes the parts of the number line where a particular condition, such as an inequality, holds true. Critical points and the behavior of the inequality around them dictate these intervals. For our problem, after analyzing the signs of the expression across the number line intervals, the solution is represented in interval notation as \([-4, 1]\).This notation indicates:

• Brackets \([]\) are used for inclusive values, meaning -4 and 1 satisfy the inequality since the inequality \(\leq 0\) allows zero.
• The terms inside the bracket, \(-4\) and \(1\), are the endpoints of the interval where the inequality holds.

Interval notation is particularly useful for showing intervals that do not include certain points—for instance, inequalities that lack \(\leq\) or \(\geq\) would use parentheses \(()\) to exclude boundary numbers.Employing interval notation simplifies the expression of solution sets, making it easier to understand and visually interpret alongside a number line representation.

Number Line

A number line is an excellent visual aid for understanding rational inequalities. It acts as a straightforward tool to display how different intervals satisfy an inequality. Before solutions are translated into interval notation, the number line helps to visualize critical numbers and their effects.For our inequality \(\frac{(x+4)(x-1)}{x+2} \leq 0\):

• The critical numbers \(-4\) and \(1\) are marked on the number line.
• These points divide the line into regions: \((-\infty, -4), (-4, 1), (1, \infty)\).
• Tests in each interval with numbers like \(-5, 0,\) and \(2\) tell us the sign of the rational expression.

By doing this, one can confirm which intervals satisfy the inequality. The number line notably highlights that the region between \(-4\) and \(1\) is where the expression assumes a non-positive value, confirming the solution \([-4, 1]\) in interval notation. The number line not only confirms findings but enhances understanding by presenting the information visually.