Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-40: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ x^{4}+15 x^{2}<16 $$

Step-by-Step Solution

Answer

The solution is \((-1, 1)\).

Step 1: Simplify the Inequality

Let us first substitute a new variable to simplify the inequality. Set \( y = x^2 \). Then the inequality becomes: \[y^2 + 15y < 16\]

Step 2: Rearrange the Equation

We move all terms to one side of the inequality to form a quadratic equation, \[y^2 + 15y - 16 < 0\]This sets us up for finding the roots of the equation.

Step 3: Use the Quadratic Formula

We apply the quadratic formula, \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 15 \), and \( c = -16 \). Calculate the discriminant: \[b^2 - 4ac = 15^2 - 4 \times 1 \times (-16) = 225 + 64 = 289\]Thus, the roots are: \[y = \frac{-15 \pm \sqrt{289}}{2}\]\[y = \frac{-15 \pm 17}{2}\]This gives the roots \( y = 1 \) and \( y = -16 \).

Step 4: Determine the Solution Interval

We already know that \( y = x^2 \) cannot be negative, so we only consider \( y = x^2 > 0 \). Thus, we consider the interval \( 0 < y < 1 \).

Step 5: Express the Solution in Terms of x

Return to the original variable \( x \), we have \( 0 < x^2 < 1 \). Therefore, the corresponding interval for \( x \) is \(-1 < x < 1\).

Step 6: Final Answer and Interval Notation

The solution to the inequality \( x^4 + 15x^2 < 16 \) is \(-1 < x < 1\). In interval notation, this is expressed as \((-1, 1)\).

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions with powers, like in our exercise with the inequality \(x^4 + 15x^2 < 16\). Solving polynomial inequalities is similar to solving equations, but with one key difference: rather than finding a singular solution, we need to find a range of values that satisfy the inequality. To tackle this problem, we transformed it by substituting \(y = x^2\), simplifying our polynomial to \(y^2 + 15y < 16\). This substitution makes it easier to manipulate and understand the polynomial's behavior.

Here are key steps in handling polynomial inequalities:

• Find roots by rearranging the inequality into a standard form, as demonstrated through \(y^2 + 15y - 16 < 0\).
• Use those roots to determine intervals that satisfy the inequality.
• Consider the nature of the polynomial, such as the fact that \(x^2\) cannot be negative, to refine your solution set.

This conceptual framework sets you up for solving more complex polynomial inequalities by breaking big problems into manageable parts.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions or "roots" of quadratic equations typically expressed in the form \(ax^2 + bx + c = 0\). In our exercise, once we substituted \(y = x^2\), we ended up transforming the polynomial inequality into the quadratic form \(y^2 + 15y - 16 < 0\).

The formula is given by:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a\), \(b\), and \(c\) are coefficients from the equation. Solving for these roots involves calculating the discriminant (\(b^2 - 4ac\)), which tells us whether the equation has real, complex, or repeated roots. Here, it's computed as 289, indicating two distinct real roots at \(y = 1\) and \(y = -16\).

Steps in using the quadratic formula:

• Identify coefficients \(a\), \(b\), and \(c\) from your equation.
• Calculate the discriminant.
• Substitute values into the quadratic formula to find the roots.

These roots play a crucial role in defining the intervals where the original inequality holds true.

Interval Notation

Interval notation is a concise way of expressing the set of solutions to an inequality. It uses brackets and parentheses to describe the starting and ending points of the intervals. In our inequality \(x^4 + 15x^2 < 16\), once we found the solution \(-1 < x < 1\), it translates into the interval notation \((-1, 1)\).

Here's a quick guide:

• Use parentheses \(()\) to indicate that the endpoint is not included in the interval, as seen in our solution.
• Use brackets \([]\) if the endpoint is included (like when you have \(x \leq 1\)).
• Combine symbols to express compound intervals, such as union \(\cup\) for multiple intervals.

This method is instrumental in communicating the solution set of inequalities clearly and efficiently. Remember, the clarity it affords is especially beneficial when dealing with more intricate expressions or multiple conditions.