Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality and write the solution set in interval notation. \(25 x^{3}-10 x^{2}<0\)

Step-by-Step Solution

Answer

\((-∞ , 0) \cup (2/5, ∞)\)

Step 1: Factorization

First, factor out the common factor from the expression. The common factor of \(25x^{3}\) and \(-10x^{2}\) is \(5x^{2}\). So, factorize the inequality: \(5x^{2}(5x - 2) < 0\)

Step 2: Find the zeros

In the next step, find the zeros (roots) of the equation equivalent to the provided inequality: \(5x^{2}(5x - 2) = 0\)Setting each factor equal to zero gives \(x = 0\) and \(x = 2/5\). These are the points where the inequality changes its sign.

Step 3: Test intervals

We have divided the number line into three intervals by the roots. These intervals are \((-∞ , 0)\), \((0, 2/5)\), and \((2/5, ∞)\). Let's pick a test point from each interval and substitute it into the original inequality to determine whether the interval is a part of the solution. Using test points as \(x = -1\), \(x = 1/4\), and \(x = 1\), we find the intervals \((-∞ , 0)\) and \((2/5, ∞)\) satisfy the inequality and \((0, 2/5)\) doesn't.

Step 4: Write the final answer in interval notation

The solution in interval notation is \((-∞ , 0) \cup (2/5, ∞)\). That represents all the \(x\) values for which the inequality holds true.

Key Concepts

Interval Notation

Interval notation is a succinct way to represent a set of real numbers that are solutions to inequalities. It allows you to convey a continuous range of values that satisfy a given condition. When using interval notation, remember these key symbols:

• Round parentheses "\(( )\)" indicate that the endpoint is not included in the interval, representing an open boundary.
• Square brackets "\([ ]\)" indicate that the endpoint is included in the interval, meaning a closed boundary.

In the example inequality \(25x^{3}-10x^{2}<0\), the solution set in interval notation is expressed as \((-∞ , 0) \cup (2/5, ∞)\). This means that for all values of \(x\) within these intervals, the original inequality holds true. The use of the union symbol "\( \cup \)" neatly combines separate intervals that form the solution set. Understanding interval notation is crucial for writing solutions to inequalities because it directly translates the number line's inclusion and exclusion of values into a compact form.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process simplifies problem-solving and allows us to identify critical points within an equation, such as roots. Taking the polynomial in the original inequality, \(25x^{3}-10x^{2}\), factoring begins by finding the greatest common factor (GCF).For \(25x^{3}\) and \(-10x^{2}\), the GCF is \(5x^{2}\). Factoring out this common factor, we rewrite the polynomial as \(5x^{2}(5x - 2)\). This expression is easier to work with as it separates the polynomial into simpler multipliers, making it manageable to solve for when it equals zero, which will help in determining sign changes.

Roots of Equations

Roots of equations refer to the \(x\)-values that make the polynomial equal to zero. In solving polynomial inequalities, finding these roots is a pivotal step. Here, setting each factor of the factored inequality \(5x^{2}(5x - 2) = 0\), gives us the roots.

• The first factor \(5x^{2} = 0\) translates to \(x = 0\).
• The second factor \(5x - 2 = 0\) simplifies to \(x = \frac{2}{5}\).

These roots divide the number line into sections, which are assessed to determine where the inequality holds.By solving \(5x^{2}(5x - 2)\) we find the points \(x = 0\) and \(x = \frac{2}{5}\), leading us to test intervals around these roots: \((-∞ , 0)\), \((0, 2/5)\), and \((2/5, ∞)\). Testing within these intervals confirms where the inequality is valid.