Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality and write the solution set in interval notation. \(4 x^{3}-x^{4} \geq 0\)

Step-by-Step Solution

Answer

The solution to the inequality \(4 x^{3}-x^{4} \geq 0\) in interval notation is (0, 4] ∪ [4, ∞).

Step 1: Factorise the Polynomial

The first step is to factorise the given polynomial. In this inequality, it can be factorised by factoring out the greatest common factor, \(x^{3}\), from each term. This results in \(x^{3}(4-x) \geq 0\)

Step 2: Determine Critical Points

Critical points are solutions when the inequality equals 0. By setting \(x^{3}\) and \(4-x\) to zero, the critical values are 0 and 4.

Step 3: Test Intervals

Create intervals using the critical numbers which will then be tested in the inequality. The intervals are (-∞, 0), (0, 4), (4, ∞). Pick a test number from each interval and substitute into the inequality. For (-∞, 0), a value of -1 can be chosen, for (0, 4), a value of 1, and for (4, ∞), a value of 5. After testing these values, the inequality holds true for intervals (0, 4) and (4, ∞) but does not hold true for interval (-∞, 0)

Step 4: Write Answer in Interval Notation

After obtaining the intervals where the inequality holds true, write the answers in interval notation as a final answer. For the given inequality, the solution set will be in the form (0, 4] ∪ [4, ∞).

Key Concepts

Polynomial Factorization

Polynomial factorization is a key technique in solving inequalities. It involves breaking a polynomial down into simpler parts, or factors, which can make the inequality easier to solve. For example, given the polynomial inequality \(4x^3 - x^4 \geq 0\), the first step is to factor the polynomial. This is done by identifying the greatest common factor (GCF) that can be extracted from each term of the polynomial. In this case, \(x^3\) is the GCF.

By factoring out \(x^3\), the polynomial simplifies to \(x^3(4-x) \geq 0\). Now, we have the polynomial in a factored form, which is simpler to work with when determining solutions.

• Benefits of Factoring: Easier identification of critical points.
• Simplifies Polynomial: Offers a clearer view of possible values.

Critical Points

Critical points are crucial in solving inequalities as they represent the values where the inequality is equal to zero or undefined. These points help break down the number line into intervals that can be tested. For the inequality \(x^3(4-x) \geq 0\), finding the critical points involves setting each factor to zero:

• Set \(x^3 = 0\), giving the critical point \(x = 0\).
• Set \(4-x = 0\), solving for \(x = 4\).

These critical points divide the number line into distinct intervals: \((-∞, 0)\), \((0, 4)\), and \((4, ∞)\). Testing values from each interval in the inequality can determine where the solution is valid.

• Purpose of Critical Points: Help in interval testing.
• Significance: Indicate changes in inequality direction.

Interval Notation

Interval notation is a concise way to express the solution set of an inequality. It uses brackets and parentheses to describe where an inequality holds true on a number line.

Once testing is done across intervals defined by critical points, we identify which intervals satisfy the inequality. Using the critical points \(x=0\) and \(x=4\), we test these intervals to determine the solution set:

• The inequality holds for intervals \((0, 4)\) and \((4, ∞)\).
• It fails in the interval \((-∞, 0)\).

Therefore, the solution in interval notation becomes \((0, 4]\cup[4, ∞)\), where the brackets indicate whether values are included or excluded.

• Round Brackets \(( )\): Indicates values are not included.
• Square Brackets \([ ]\): Indicates values are included.