Problem 46 - Algebra and Trigonometry

Question

\(33-66\) . 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 set is \((-\infty, -3] \cup [3, \infty)\).

Step 1: Set Up the Inequality

We are given the inequality \( x^2 \geq 9 \). To find the solution, we first need to write this as two inequalities since the expression involves squaring.

Step 2: Solve for Critical Points

Solve the equation \( x^2 = 9 \) to find the critical points. This can be done by taking the square root of both sides, giving us \( x = 3 \) and \( x = -3 \). These points will be used to determine the intervals to check.

Step 3: Test Intervals Around Critical Points

The critical points divide the number line into three intervals: \((-\infty, -3)\), \([-3, 3]\), and \((3, \infty)\). We need to test points from each interval in the original inequality \(x^2 \geq 9\) to see where it holds true.

Step 4: Check Each Interval

- For \( x > 3 \), try \( x = 4 \), which gives \( 4^2 = 16 \geq 9 \), so this interval works.- For \( x = -3 \) and \( x = 3 \), the test points of \( -3 \) and \( 3 \) satisfy \( (-3)^2 = 9 \) and \( 3^2 = 9 \), so these points are included.- For \( -3 < x < 3 \), try \( x = 0 \), which gives \( 0^2 = 0 ot\geq 9 \), so this interval does not work.

Step 5: Write the Solution in Interval Notation

From the tests, we see the solution is where \( x \leq -3 \) or \( x \geq 3 \), corresponding to the union of the intervals \(( -\infty, -3] \cup [3, \infty )\).

Step 6: Graph the Solution

On a number line, represent the solution with a closed circle at \(-3\) and \(3\), and shade all values less than or equal to \(-3\) and greater than or equal to \(3\). This indicates the intervals \(( -\infty, -3] \) and \([3, \infty )\) are included in the solution.

Key Concepts

Quadratic Inequalities

Quadratic inequalities are a crucial part of algebra and involve expressions with a squared variable. Unlike simple linear inequalities, quadratic inequalities can present solutions involving two or more intervals on the number line. Understanding these inequalities requires a grasp of the quadratic equation. A typical form might look like this: \[ ax^2 + bx + c \geq 0 \] or the simpler \[ x^2 \geq 9 \].
To solve them, one often starts by considering the related quadratic equation. For example, to solve \[ x^2 \geq 9 \], we first solve the equation \[ x^2 = 9 \] to find our critical points. By determining these points, you can help establish the boundaries of intervals you'll test to determine where the inequality holds true. Quadratic inequalities often provide solution sets that are unions of intervals, revealing where the inequality is valid.

Interval Notation

Interval notation is a concise way to express ranges of numbers, which are solutions to inequalities. It's especially handy with quadratic inequalities, where solutions often encompass multiple intervals. For instance, when solving \[ x^2 \geq 9 \], we found the critical points at \[ x = -3 \] and \[ x = 3 \]. This gives us intervals we need to explore.

• "(a, b)" indicates all numbers between "a" and "b", without including "a" and "b" themselves.
• "[a, b]" includes both "a" and "b" in the interval.

In interval notation, we express the solution, such as ((-\[\infty\],-3] \cup [3,\[\infty\])) , to clearly convey where the inequality holds.
The bracket type you choose is crucial as it indicates whether or not the endpoints are included. In the solution above, \([-3, 3]\) includes the endpoints because the inequality involves \(\geq\) which specifies inclusion.

Critical Points

Critical points are key values where the quadratic expression either changes sign or equals zero. They play a vital role in breaking down the solution of quadratic inequalities. Finding these points involves setting the quadratic expression equal to zero and solving for the variables.
In our given inequality \(x^2 \geq 9\), we find the critical points by setting \(x^2 = 9\). Solving this equation results in two critical points: \(-3\) and \(3\). These points divide the number line into distinct intervals that need evaluation.
Once we know the critical points, we can test the intervals between and around them to determine where the original inequality holds. This testing involves selecting test points from each interval. If a test point satisfies the inequality, the entire interval does.

Solution Sets

A solution set is the collection of all values that satisfy an inequality. For quadratic inequalities, these sets often consist of multiple intervals on the number line. In the example of \(x^2 \geq 9\), after identifying critical points and testing intervals, we observe which sections of the number line fulfill the inequality.
The solution involves values both less than or equal to \(-3\) and greater than or equal to \(3\). Thus, the solution set is given as ((-\[\infty\],-3] \cup [3,\[\infty\])).
Visually, this set can be depicted on a number line, with closed circles at \(-3\) and \(3\) to show endpoint inclusion, and shaded lines extending infinitely in both negative and positive directions from these points.
Understanding solution sets not only aids in solving the inequalities but also enhances comprehension of what the answers represent in practical terms.