Problem 54 - Algebra and Trigonometry

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}\left(x^{2}-1\right) \geq 0 $$

Step-by-Step Solution

Answer

Solution: \((-\infty, -1] \cup \{0\} \cup [1, \infty)\).

Step 1: Identify the Roots

To solve the inequality \(x^2(x^2 - 1) \geq 0\), we first find the roots of the equation \(x^2(x^2 - 1) = 0\). \(x^2 = 0\) gives \(x = 0\), and \(x^2 - 1 = 0\) gives \(x = \pm 1\). Thus, the roots of the equation are \(x = -1, 0, 1\).

Step 2: Determine the Test Intervals

Using the roots, split the number line into intervals: \((-\infty, -1)\), \([-1, 0)\), \((0, 1)\), and \((1, \infty)\). We need to test values from each interval to determine where the inequality holds true.

Step 3: Test Each Interval

Select a test point from each interval:- For \((-\infty, -1)\), use \(x = -2\): \((-2)^2((-2)^2 - 1) = 4 \times 3 = 12 \geq 0\) (True).- For \([-1, 0)\), use \(x = -0.5\): \((-0.5)^2((-0.5)^2 - 1) = 0.25 \times -0.75 = -0.1875 \geq 0\) (False).- For \((0, 1)\), use \(x = 0.5\): \((0.5)^2((0.5)^2 - 1) = 0.25 \times -0.75 = -0.1875 \geq 0\) (False).- For \((1, \infty)\), use \(x = 2\): \((2)^2((2)^2 - 1) = 4 \times 3 = 12 \geq 0\) (True).

Step 4: Check Boundary Points

Since \(x = -1, 0, 1\) are roots, evaluate these points:- At \(x = -1\): \((-1)^2((-1)^2 - 1) = 1 \times 0 = 0\) (True).- At \(x = 0\): \(0^2(0^2 - 1) = 0 \times -1 = 0\) (True).- At \(x = 1\): \(1^2(1^2 - 1) = 1 \times 0 = 0\) (True).

Step 5: Write the Solution in Interval Notation

The solution set includes intervals where the inequality is true: \((-\infty, -1] \cup \{0\} \cup [1, \infty)\).

Key Concepts

Interval Notation

When we solve inequalities, we often present the solution using a format known as interval notation. This is a way to describe a set of numbers that are included within certain boundaries.
For instance, if a solution includes all numbers between -1 and 0, we can write this as \((-1, 0)\). Here, parentheses \(( )\) indicate that the endpoints are not included.
If boundaries should be included, such as all numbers between and including -1 and 0, we use brackets \([ ]\) to indicate this: \([-1, 0]\).
In the current problem, the solution was given as \((-\infty, -1] \cup \{0\} \cup [1, \infty)\). This notation tells us:

• All numbers less than and including -1.
• The number 0 is also a part of the solution set.
• All numbers greater than and including 1.

This concise format helps us quickly understand which values satisfy the inequality.

Roots of Equations

The roots of an equation are the solutions where the expression equals zero. Finding these roots is often the first step in solving inequalities.
Consider the equation \(x^2(x^2 - 1) = 0\). To find the roots:

• Start by setting each factor to zero.
• For \(x^2 = 0\), solve to find that \(x = 0\).
• For \(x^2 - 1 = 0\), factor to \((x - 1)(x + 1) = 0\), giving roots \(x = 1\) and \(x = -1\).

The roots \(-1, 0, 1\) divide the number line into intervals, which we can test to find where the inequality holds true.
These points also inform us about the critical values, where the inequality shifts from true to false or vice versa.

Test Intervals

Once the roots of the equation are identified, we divide the number line into intervals based on these roots. This allows us to see which intervals make the inequality true.
Using the roots \(-1, 0, 1\), the number line is divided into these intervals:

• \((-\infty, -1)\)
• \([-1, 0)\)
• \((0, 1)\)
• \([1, \infty)\)

Each of these ranges needs testing with a sample point inside the interval.
For example, choose \(x = 2\) for the interval \((1, \infty)\). Plugging \(x = 2\) into the expression returns a positive result, so the inequality holds true in that interval. This testing tells us where the inequality is satisfied, a vital part of fully solving the problem.

Graphing Inequalities

Graphing inequalities helps visualize the solution set on the number line. This visual aid makes complex solutions more understandable.
Once you've determined which intervals satisfy the inequality, graph these ranges as follows:

• Draw a line representing the number line.
• Mark the roots \(-1, 0,\) and \(1\) as points of interest.
• For intervals where the inequality is true using \((-\infty, -1]\) and \([1, \infty)\), shade the entire line and solid circles at -1 and 1 to include these points.
• Place a dot at 0 to highlight it's a single point where the inequality is zero and including it as part of the solution set.

By shading or marking these intervals appropriately, you create a clear picture of where the inequality holds.
Graphing is a powerful tool to confirm the algebraic approach and ensure the solution makes sense.