Problem 48 - Precalculus Mathematics for Calculus

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$x^{2} \geq 9$$

Step-by-Step Solution

Answer

The solution is \((-\infty, -3] \cup [3, \infty)\).

Step 1: Rearrange the Inequality

First, rewrite the inequality in a more standard form: \(x^2 - 9 \geq 0\). This is equivalent to \(x^2 \geq 9\).

Step 2: Factor the Expression

Recognize that the left side of the inequality is a difference of squares. This can be factored as \((x - 3)(x + 3) \geq 0\).

Step 3: Identify Critical Points

Set each factor equal to zero to find the critical points: \(x - 3 = 0\) gives \(x = 3\), and \(x + 3 = 0\) gives \(x = -3\).

Step 4: Test Intervals and Signs

Divide the number line into intervals based on the critical points: - Interval 1: \((-\infty, -3)\)- Interval 2: \((-3, 3)\)- Interval 3: \((3, \infty)\)Test a point from each interval to determine if the expression \((x-3)(x+3)\) is positive or negative.- For Interval 1 (e.g., \(x = -4\)): \((-4 - 3)(-4 + 3) = 7\) (positive)- For Interval 2 (e.g., \(x = 0\)): \((0 - 3)(0 + 3) = -9\) (negative)- For Interval 3 (e.g., \(x = 4\)): \((4 - 3)(4 + 3) = 7\) (positive)

Step 5: Include Boundary Points

The inequality \((x - 3)(x + 3) \geq 0\) means we include points where it equals zero. This includes the critical points \(-3\) and \(3\).

Step 6: Write Solution in Interval Notation

Since the expression is positive on intervals \((-\infty, -3)\) and \((3, \infty)\), and zero at the points \(-3\) and \(3\), the solution in interval notation is \((-\infty, -3] \cup [3, \infty)\).

Key Concepts

Nonlinear Inequalities

When dealing with inequalities like \(x^2 \geq 9\), it's important to recognize that they are not straightforward like linear inequalities. Nonlinear inequalities involve expressions that are not just lines, but curves such as parabolas, circles, or other shapes.

In our exercise, the expression involves a square term \(x^2\), which forms a parabola when graphed. Nonlinear inequalities can be more complex to solve, but the fundamental approach remains similar:

• Rearrange the inequality to a standard form similar to an equation.
• Factor the expression, if possible, to find where the expression changes its sign.
• Identify critical points where the parabolic curve crosses the x-axis.
• Check each interval between these points separately to determine where the inequality holds true.

Interval Notation

Interval notation is a way to describe a set of numbers along a number line. It's particularly helpful when expressing the solution of inequalities. The main symbols used in interval notation are:

• Parentheses \(( \) and \()\) indicate that an endpoint is not included in the interval, also known as open intervals.
• Brackets \([ \) and \()]\) signify that an endpoint is included, or a closed interval.
• The symbol \(\cup\) represents a union of intervals, combining separate solutions into a single expression.

In the solution \(( -\infty, -3] \cup [3, \infty)\), the union of two intervals shows where the inequality \(x^2 \geq 9\) holds true. Here, -3 and 3 are included in the solution, marked by the closed brackets, while the intervals continue infinitely in either direction, represented by \(-\infty\) and \(\infty\).

Critical Points

Critical points are values that separate different behaviors of a function or expression. In the context of inequalities, critical points are where the expression equals zero. They are crucial because they divide the number line into intervals where the inequality may switch signs.

For the inequality \(x^2 - 9 \geq 0\), factoring it into \((x-3)(x+3) \geq 0\) reveals the critical points by solving \(x-3=0\) and \(x+3=0\). These solutions, \(x=3\) and \(x=-3\), are the points where the expression equals zero.

• Critical points help to determine intervals where the expression is either positive or negative.
• Each interval created by critical points needs to be tested to see if it satisfies the inequality.
• These points also help in understanding where the curve touches or crosses the x-axis.

Factoring

Factoring is a technique used to simplify expressions and is essential in solving nonlinear inequalities. The idea is to express a polynomial as a product of its factors, making it easier to identify critical points.

In the given problem \(x^2 - 9 \geq 0\), recognizing it as a difference of squares allows us to factor it into \((x-3)(x+3)\). This step transforms the inequality into a product of binomials, making the solution process more manageable.

• Factoring simplifies complex expressions by breaking them into simpler parts.
• It helps in locating the critical points where the expression equals zero.
• Understanding factoring is crucial for dealing with various types of polynomial inequalities.

By factoring, we can quickly translate the inequality into an accessible format, setting the stage for testing intervals and finding solutions in simple and understandable steps.