Problem 45 - College Algebra with Modeling and Visualization

Question

Solve the polynomial inequality (a) symbolically and (b) graphically. $$ x^{4}-13 x^{2}+36<0 $$

Step-by-Step Solution

Answer

The solution is \(-3 < x < -2\) or \(2 < x < 3\).

Step 1: Rewrite the Inequality

First, we'll rewrite the inequality in terms of a factored expression. The given inequality is \( x^4 - 13x^2 + 36 < 0 \). Notice that \( x^4 = (x^2)^2 \), so we can treat this as a quadratic in terms of \( x^2 \).

Step 2: Factor the Expression

Let's set \( y = x^2 \), so our inequality becomes \( y^2 - 13y + 36 < 0 \). Factoring the quadratic, we find: \( (y - 4)(y - 9) < 0 \). Therefore, \( x^4 - 13x^2 + 36 = (x^2 - 4)(x^2 - 9) < 0 \).

Step 3: Solve for Critical Points

Find the values of \( x \) that make \( (x^2 - 4) \) or \( (x^2 - 9) \) zero. These occur at \( x^2 = 4 \) and \( x^2 = 9 \). Thus, \( x = \pm 2 \) and \( x = \pm 3 \). These are the critical points.

Step 4: Test Intervals around Critical Points

The critical points divide the number line into intervals: \((-\infty, -3)\), \((-3, -2)\), \((-2, 2)\), \((2, 3)\), and \((3, \infty)\). Choose test points in these intervals to see where the inequality holds. Compute the sign of \((x^2 - 4)(x^2 - 9)\) in each interval.

Step 5: Analyze Interval Signs

- For \((-\infty, -3)\), test \(x = -4\): both \(x^2 - 4\) and \(x^2 - 9\) are positive, so the product is positive. - For \((-3, -2)\), test \(x = -2.5\): \(x^2 - 4\) is positive, \(x^2 - 9\) is negative, so the product is negative. - For \((-2, 2)\), test \(x = 0\): both are negative, so the product is positive.- For \((2, 3)\), test \(x = 2.5\): \(x^2 - 4\) is positive, \(x^2 - 9\) is negative, so the product is negative.- For \((3, \infty)\), test \(x = 4\): both are positive, so the product is positive.

Step 6: Determine Solution Set

Based on the interval test results, the inequality \( (x^2 - 4)(x^2 - 9) < 0 \) is satisfied in the intervals \((-3, -2)\) and \((2, 3)\). Therefore, the solution set symbolically is \(-3 < x < -2 \) or \(2 < x < 3\).

Step 7: Verify Graphically

Graphically, plot \( y = x^4 - 13x^2 + 36 \). The graph will intersect the x-axis at the critical points \( x = \pm 2 \) and \( x = \pm 3 \). The regions where the graph is below the x-axis will correspond to \( x \) values that satisfy the inequality. From the graph, you should confirm that the inequality holds true between \(-3 < x < -2\) and \(2 < x < 3\) as determined symbolically.

Key Concepts

Critical Points

To solve polynomial inequalities, understanding critical points is essential. Critical points are values of the variable where the polynomial equals zero or has undefined behavior. For the inequality \( x^4 - 13x^2 + 36 < 0 \), we start by identifying the expressions \( x^2 - 4 \) and \( x^2 - 9 \).
To find the critical points, set each expression equal to zero:

• \( x^2 - 4 = 0 \) gives \( x = \pm 2 \)
• \( x^2 - 9 = 0 \) gives \( x = \pm 3 \)

These values \( x = -3, -2, 2, \text{ and } 3 \) divide our number line into different regions. Critical points mark the boundaries where the sign of the polynomial may change, making them crucial in determining where the inequality holds true.

Interval Testing

Interval testing is a powerful technique used to determine where an inequality is true. After identifying the critical points, the next task is to test the signs of the polynomial in the intervals defined by these points.

The intervals divided by the critical points are: \((-\infty, -3)\), \((-3, -2)\), \((-2, 2)\), \((2, 3)\), and \((3, \infty)\).

To test these intervals:

• Select a test point from each interval.
• Plug these test points into the factored form \((x^2 - 4)(x^2 - 9)\).
• Determine if the product is positive or negative.

For example, for \((-3, -2)\) testing with \(x = -2.5\), we find that \( x^2 - 4 > 0 \) and \( x^2 - 9 < 0 \), making the product negative. Conduct similar checks for other intervals.

Factoring Quadratics

Factoring is a fundamental part of solving polynomial inequalities. In our problem, the expression to factor is \( x^4 - 13x^2 + 36 \).

Recognizing that \( x^4 \) can be rewritten as \((x^2)^2\), we let \( y = x^2 \). This transforms the polynomial into a quadratic in terms of \( y \):

• \( y^2 - 13y + 36 \).

Next, we factor the quadratic into:

• \( (y - 4)(y - 9) \).

Substituting \( y = x^2 \) back gives us:

• \( (x^2 - 4)(x^2 - 9) \).

This factorization is critical because it allows the application of the zero product property to find the critical points and solve the inequality.

Graphical Solution

A graphical solution provides a visual interpretation of the inequality. Plotting the graph of the function \( y = x^4 - 13x^2 + 36 \) helps confirm symbolic results.

When graphing, pay attention to:

• Intersection points at \( x = \pm 2 \) and \( x = \pm 3 \) where the curve crosses the x-axis.
• The shape of the curve, showing how the function moves from positive to negative and vice versa.

In our problem, the segments where the graph lies below the x-axis represent solution intervals for the inequality. Upon observing the graph, it confirms that the inequality holds between \(-3 < x < -2\) and \(2 < x < 3\), aligning with the interval testing results. Graphs can often reveal trends and characteristics not immediately evident in symbolic analysis, making them a valuable tool in solving polynomial inequalities.