Question

For which of the following intervals of \(x\) is \(x^{2}>\frac{1}{x^{2}}\) ? (1) \((-\infty,-1) \cup(1, \infty)\) (2) \((-\infty,-1) \cup(1, \infty)\) (3) \((-1,1)\) (4) \([-1,1]\)

Short answer

Answer: The inequality holds true for \(x \in (-\infty, -1) \cup (1, \infty)\).

Step-by-step solution

Remove the fractions from the inequality

To simplify the inequality, we will multiply both sides by \(x^2\). However, since \(x^2\) can be negative, positive, or zero, we need to consider the three possible cases separately.

Case 1 (\(x^2 > 0\))

If \(x^2 > 0\), then we can multiply both sides by \(x^2\) without changing the direction of the inequality. In this case, we get: \(x^2 \cdot x^2 > \frac{1}{x^2} \cdot x^2\) which simplifies to: \(x^4 > 1\) In this case, x can be in the intervals \((-\infty, -1)\) or \((1, \infty)\).

Case 2 (\(x^2 < 0\))

Since \(x^2\) is always greater than or equal to zero, there is no such case where \(x^2 < 0\). Therefore, we don't need to consider this case.

Case 3 (\(x^2 = 0\))

If \(x^2 = 0\), then \(x = 0\). In this case, the inequality becomes: \(0 > \frac{1}{0^2}\) Since division by zero is undefined, this case does not provide a valid solution to the inequality.

Combine the solutions from all three cases

Since there were no valid solutions in Case 2 and Case 3, the final solution of the inequality is the set of intervals found in Case 1: \(x \in (-\infty, -1) \cup (1, \infty)\) Comparing our solution with the given answer options, we can conclude that the correct answer is option (1).

Key concepts

Quadratic inequalities

In mathematics, quadratic inequalities involve terms with a variable raised to the power of two, such as in the expression \(x^2 > \frac{1}{x^2}\). Solving these types of inequalities is an important skill that allows us to determine ranges of values for which the inequality holds true. Here’s how to approach quadratic inequalities:

• Begin by analyzing the inequality to understand its structure. In our example, \(x^2 > \frac{1}{x^2}\) involves a straightforward inequality between a quadratic term and a fraction.
• The first step in simplifying such an inequality is to eliminate any fractions. We achieve this by multiplying both sides by the variable's square, \(x^2\), noting that \(x^2\) must be balanced carefully because its sign depends on \(x\).
• Once the fractions are removed, the inequality \(x^4 > 1\) becomes apparent and more manageable to solve.

Addressing quadratic inequalities often requires splitting the problem into cases based on the value of \(x^2\), such as positive, zero, or negative. This ensures clarity, as seen in the full solution.

Interval notation

Interval notation is a concise way of representing sets of numbers, often used with inequalities to express solutions. Understanding interval notation is crucial for clearly communicating the range of solutions derived from inequalities like \(x^4 > 1\).

• An interval refers to a set of numbers lying between two endpoints. For instance, the interval \((1, \infty)\) includes all numbers greater than 1.
• Parentheses \(()\) indicate that endpoints are not included, whereas square brackets \([]\) mean they are included.
• For the inequality \(x^4 > 1\), interval notation helps articulate the solution set as \((-\infty, -1) \cup (1, \infty)\), which describes all \(x\) where quadratic conditions hold, except the endpoints are excluded.

Using interval notation to communicate the solution provides a clear and precise mathematical expression of the possible values of \(x\) within the context of the inequality.

Solving inequalities

Solving inequalities involves finding which values of a variable satisfy a given condition. Unlike equations, which are solved for exact values, inequalities define a range of possibilities. Here's a guide to tackling inequalities:

• First, simplify the inequality by performing arithmetic operations to both sides. This might involve multiplying, dividing, or adding terms across the inequality to make it easier to solve.
• Bear in mind the properties of inequalities: multiplying or dividing both sides by a negative number reverses the inequality’s direction. This detail is crucial when managing cases involving variables, as seen with \(x^2\) in our problem.
• Once the inequality is simplified, identify critical values that may serve as boundaries in a solution set, such as the solutions of \(x^4 = 1\). These values divide the number line into intervals to test.
• Test values within each interval to see if they satisfy the original inequality. This step ensures that you include the appropriate ranges in your final solution.

Through logical steps and careful considerations, you can skillfully resolve inequalities and confidently report solutions using methods like interval notation.