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Chapter 1: Problem 50

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

Short Answer

Expert verified
The solution is \((-\infty, -3] \cup [3, \infty)\) and is graphically represented by shading those intervals on the number line.

Step by step solution

01

Identify the Inequality

We are given the inequality \( x^2 \geq 9 \). This inequality is nonlinear because the highest power of \( x \) is 2.
02

Rewrite as Equation for Critical Points

Set \( x^2 = 9 \) to find critical points. Solving gives \( x = 3 \) or \( x = -3 \). These points divide the number line into intervals to test.
03

Test Intervals

The critical points \( x = -3 \) and \( x = 3 \) create three intervals: \((-fty, -3)\), \([-3, 3](-3, 3)\), and \((3, fty)\). We will test a point from each interval to see if it satisfies \( x^2 \geq 9 \).
04

Test Point in \((-\infty, -3)\)

Choose \( x = -4 \). Substitute into \( x^2 \geq 9 \): \((-4)^2 = 16 \geq 9 \). The inequality is satisfied.
05

Test Point in \((-3, 3)\)

Choose \( x = 0 \). Substitute into \( x^2 \geq 9 \): \(0^2 = 0 ot\geq 9 \). The inequality is not satisfied.
06

Test Point in \((3, \infty)\)

Choose \( x = 4 \). Substitute into \( x^2 \geq 9 \): \(4^2 = 16 \geq 9 \). The inequality is satisfied.
07

Verify Boundary Points

Check \( x = -3 \) and \( x = 3 \): \((-3)^2 = 9 \text{ and } 3^2 = 9 \), so both satisfy \( x^2 \geq 9 \). Thus, \(-3\) and \(3\) are included.
08

Write Solution in Interval Notation

Since the intervals \((-\infty, -3] \) and \([3, \infty)\) satisfy the inequality and include the endpoints, the solution is \((-\infty, -3] \cup [3, \infty) \).
09

Graph the Solution Set

Draw a number line, with closed circles at -3 and 3. Shade the regions \((-\infty, -3] \) and \([3, \infty) \) to indicate they are part of the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequalities
Inequalities are expressions that compare two values or expressions, indicating that one is larger, smaller, or not equal to the other. In mathematics, inequalities play a crucial role in identifying ranges of solutions rather than fixed numbers. Inequalities are often expressed using symbols such as:
  • \(<\)
  • \(>\)
  • \(\leq\)
  • \(\geq\)
In the example of the nonlinear inequality \(x^2 \geq 9\), we deal with the symbol \(\geq\), which implies that \(x^2\) is greater than or equal to 9. Unlike equations, inequalities can have a range of solutions, and finding the specific solution sets requires identifying intervals where these conditions hold true.
Interval Notation
Interval notation is a concise way of writing the set of numbers that form the solution to an inequality. It focuses on the starting and ending points of an interval, and often includes symbols to indicate whether the endpoints are included or omitted. The symbols used are:
  • "[" or "]" indicates the endpoint is included (closed interval).
  • "(" or ")" indicates the endpoint is not included (open interval).
For the inequality \(x^2 \geq 9\), the solution involves the intervals \(( -\infty, -3 ]\) and \([ 3, \infty )\). Here, -3 is included in the interval \(( -\infty, -3 ]\) because at \(x = -3\), the expression \(x^2\) equals 9. Similarly, 3 is included in the interval \([ 3, \infty )\). This notation allows us to easily express these ranges as a part of the solution.
Critical Points
Critical points are specific values that make an equation equal to zero or a fixed number and help in dividing the inequality into testable intervals. When solving inequalities like \(x^2 \geq 9\), determining the critical points is essential. To find the critical points, replace the inequality sign with an equal sign: \(x^2 = 9\). Solving gives us \(x = 3\) and \(x = -3\). These points represent the boundaries between which the sign of the inequality could change. By evaluating points within each segment defined by these critical points, you can determine where the inequality holds true.
Solution Set
The solution set of an inequality is the complete range of values that satisfy the inequality. For nonlinear inequalities, this involves not only the correct placement of interval endpoints but also the consideration of the inequality's behavior between and at these endpoints.In the inequality \(x^2 \geq 9\), the solution set consists of the numbers where the inequality holds. Testing intervals around the critical points \(-3\) and \(3\) helps verify this.
  • In the interval \((-\infty, -3)\), substituting a test point such as \(x = -4\) confirms that \((-4)^2 = 16\) satisfies \(x^2 \geq 9\).
  • In the interval \( (-3, 3)\), a test point like \(x = 0\) shows \(0^2 = 0\), which does not satisfy the inequality \(0 ot\geq 9\).
  • In the interval \((3, \infty)\), using a test point, for example, \(x = 4\), shows that \(4^2 = 16\) satisfies \(x^2 \geq 9\).
Thus, the solution set is given by the union of intervals \(( -\infty, -3 ]\) and \([ 3, \infty )\). This is expressed succinctly using interval notation, helping to communicate the range of solutions effectively.

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