Problem 37 - Precalculus Mathematics for Calculus

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$x(2 x+7) \geq 0$$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{7}{2}] \cup [0, \infty)\).

Step 1: Find the Critical Points

To solve the inequality \(x(2x + 7) \geq 0\), first identify the critical points by setting each factor equal to zero. Solving \(x = 0\) and \(2x + 7 = 0\) gives us the critical points. Solving \(2x + 7 = 0\), we get \(2x = -7\), so \(x = -\frac{7}{2}\). Thus, the critical points are \(x = 0\) and \(x = -\frac{7}{2}\).

Step 2: Interval Testing

Use the critical points to divide the number line into intervals: \((-\infty, -\frac{7}{2})\), \((-\frac{7}{2}, 0)\), and \((0, \infty)\). Test a point from each interval in the inequality \(x(2x + 7) \geq 0\): - For \((-\infty, -\frac{7}{2})\), test \(x = -4\): \((-4)(2(-4) + 7) = -4(-8 + 7) = -4(-1) = 4\), which is positive.- For \((-\frac{7}{2}, 0)\), test \(x = -1\): \((-1)(2(-1) + 7) = -1(-2 + 7) = -1(5) = -5\), which is negative.- For \((0, \infty)\), test \(x = 1\): \((1)(2(1) + 7) = 1(2 + 7) = 9\), which is positive.

Step 3: Construct the Solution Set

Since the original inequality is \(x(2x + 7) \geq 0\), we include the intervals where the expression is positive or zero. From interval testing, the solution includes \((-\infty, -\frac{7}{2}]\) and \([0, \infty)\). Since \(x = -\frac{7}{2}\) and \(x = 0\) make the expression zero, they are included in the solution set.

Step 4: Express in Interval Notation and Graph

The solution in interval notation is \((-\infty, -\frac{7}{2}] \cup [0, \infty)\). To graph the solution set, draw a number line. For the interval \((-\infty, -\frac{7}{2}]\), use a shaded line moving left with a closed circle at \(-\frac{7}{2}\). For \([0, \infty)\), use a shaded line moving right from 0 with a closed circle at 0.

Key Concepts

Critical Points

In a nonlinear inequality like the one given \[ x(2x + 7) \geq 0 \], finding the critical points is the essential first step. Critical points are where the factors of the equation are zero because these points determine the changes in the inequality's direction. In simpler terms, a critical point is a value for the variable where the mathematical expression equals zero. To find these points, set each factor in the inequality to zero and solve for \( x \).

• For \( x(2x + 7) = 0 \), break it down to individual factors.
• First, \( x = 0 \) is one critical point.
• Next, solve \( 2x + 7 = 0 \) leading to \( 2x = -7 \) and simplify to find \( x = -\frac{7}{2} \).

Therefore, the critical points at this stage of the problem are \( x = 0 \) and \( x = -\frac{7}{2} \). These are crucial for determining the intervals on the number line where the expression will be tested.

Interval Notation

After identifying critical points, the next task is to analyze the inequality on the number line using these points. The critical points help partition the number line into several segments or intervals. These intervals are tested to determine the nature (positive or negative) of the inequality within each.

• The intervals are evaluated as follows:
  1. \((-\infty, -\frac{7}{2})\)
  2. \((-\frac{7}{2}, 0)\)
  3. \((0, \infty)\)

To express the result using interval notation:

• The intervals where the inequality holds true (including equals zero) are combined.
• Here, values from \((-\infty, -\frac{7}{2}]\) and \([0, \infty)\) are analyzed.

Notice that the brackets \(( ])\) indicate whether endpoints are included in the solution.

Solution Set

A solution set is a collection of all possible values of \( x \) that satisfy the inequality. Here, the goal was determining where the expression \( x(2x + 7) \geq 0 \) is non-negative.
Testing points from different intervals sorted out which ranges satisfy the inequality:

• On testing, \( x = -4 \) in \((-\infty, -\frac{7}{2})\) gives positive output, while \( x = -1 \) in the middle interval \((-\frac{7}{2}, 0)\) gives negative output.
• The final interval, starting from zero, includes a positive value \( x = 1 \).

Therefore, the solution set can be written as \[ (-\infty, -\frac{7}{2}] \cup [0, \infty) \].
These intervals represent the x-values where the inequality holds true. On a graph, closed circles are drawn on endpoints \( x = -\frac{7}{2} \) and \( x = 0 \) showing they are part of the solution.