Problem 45 - Algebra for College Students

Question

Solve each inequality. $$3 x^{3}+12 x^{2}>0$$

Step-by-Step Solution

Answer

The solution is \((-4, 0) \cup (0, \infty)\).

Step 1: Factor the Inequality

The inequality given is \(3x^3 + 12x^2 > 0\). Notice that both terms have a common factor, \(3x^2\). We factor this out:\[3x^2(x + 4) > 0\].

Step 2: Identify Critical Points

To determine where the expression \(3x^2(x+4)\) is equal to zero, find the values of \(x\) where each factor is zero:1. \(3x^2 = 0\) gives \(x = 0\).2. \(x + 4 = 0\) gives \(x = -4\).Thus, the critical points are \(x = 0\) and \(x = -4\).

Step 3: Test Intervals Between Critical Points

The critical points divide the number line into intervals: 1. \((-\infty, -4)\)2. \((-4, 0)\)3. \((0, \infty)\)Choose test points in each interval to check the sign of the expression \(3x^2(x + 4)\):- For \((-\infty, -4)\), use \(x = -5\): \(3(-5)^2(-5 + 4) = 75 \times -1 = -75\) (negative)- For \((-4, 0)\), use \(x = -1\): \(3(-1)^2(-1 + 4) = 3 \times 3 = 9\) (positive)- For \((0, \infty)\), use \(x = 1\): \(3(1)^2(1 + 4) = 3 \times 5 = 15\) (positive).

Step 4: Determine Solution Set

From testing the intervals, we see that the expression \(3x^2(x + 4) > 0\) on the intervals \((-4, 0)\) and \((0, \infty)\). Thus, the solution to the inequality \(3x^3 + 12x^2 > 0\) is the union of these intervals:\((-4, 0) \cup (0, \infty)\).

Key Concepts

Factoring

Factoring is a technique used in algebra to simplify expressions and solve equations or inequalities. In the given inequality, \(3x^3 + 12x^2 > 0\), factoring assists in breaking down the expression into a simpler form. Let's look at this process step-by-step.The first thing you notice is that both terms, \(3x^3\) and \(12x^2\), share a common factor of \(3x^2\). By factoring out \(3x^2\), the expression becomes \(3x^2(x + 4)\). This helps to simplify the inequality, making it easier to identify the values of \(x\) for which the expression holds true.Factoring plays an essential role because it breaks down the problem, showing us critical points in the inequality we wouldn't easily see otherwise. Once factored, it becomes straightforward to work with and solve further.

Critical Points

Critical points are values of \(x\) that make the expression zero or undefined. In the context of inequalities, these points are important because they help us determine where the expression changes sign. For the inequality \(3x^2(x + 4) > 0\), we first find the critical points by setting each factor equal to zero.- The first factor, \(3x^2 = 0\), gives \(x = 0\).- The second factor, \(x + 4 = 0\), provides another critical point \(x = -4\).Identifying critical points is crucial as they segment the number line into different intervals. Each interval may have a different sign, influencing the solution.

Number Line

Once critical points are identified, we plot them on a number line. The points \(x = 0\) and \(x = -4\) divide the number line into three distinct intervals:

• Interval 1: \((-\infty, -4)\)
• Interval 2: \((-4, 0)\)
• Interval 3: \((0, \infty)\)

Using the number line helps us visualize how the sign of the expression changes across these intervals. By understanding the number line divisions, we can choose appropriate test points to determine whether the inequality holds within each section. This visual representation makes the problem-solving process more intuitive and comprehensive.

Interval Testing

Interval testing is the strategy used to determine the sign of an expression over each segment marked by critical points on the number line. With the inequality \(3x^2(x + 4) > 0\), we test each interval to see where the expression holds true.- For the interval \((-\infty, -4)\), choose \(x = -5\). Substitute back to find that the expression is negative.- In \((-4, 0)\), use \(x = -1\). Here, the expression evaluates to a positive result.- For \((0, \infty)\), select \(x = 1\). Substituting yields another positive result.By examining these results, we can conclude that the inequality is satisfied for the intervals \((-4, 0)\) and \((0, \infty)\). Interval testing is a powerful tool in analyzing inequalities, helping us pinpoint the exact solution set where the inequality is valid.