Problem 30 - Intermediate Algebra

Question

Solve each inequality. $$ 2 x^{2}-18 \geq 0 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Quadratic Expression

The inequality we need to solve is \(2x^2 - 18 \geq 0\). This is a quadratic inequality with the quadratic expression \(2x^2 - 18\).

Step 2: Set the Quadratic Equal to Zero

To find the critical points that may change the sign of the inequality, we first solve for where the expression is equal to zero. Set \(2x^2 - 18 = 0\).

Step 3: Solve for the Critical Points

Factor or use algebraic manipulation to solve the equation: \(2x^2 = 18\) becomes \(x^2 = 9\). Taking the square root gives the solutions \(x = 3\) and \(x = -3\).

Step 4: Determine the Sign Intervals

The solutions \(x = -3\) and \(x = 3\) divide the number line into three intervals: \((-\infty, -3)\), \([-3, 3]\), and \((3, \infty)\). We need to test each interval to determine where the inequality \(2x^2 - 18 \geq 0\) holds true.

Step 5: Test the Intervals

Check a test point from each interval in the inequality:\- For \((-\infty, -3)\), use \(x = -4\): \((2(-4)^2 - 18 = 14) \geq 0\) holds true.\- For \([-3, 3]\), use \(x = 0\): \((2(0)^2 - 18 = -18) ot\geq 0\) does not hold.\- For \((3, \infty)\), use \(x = 4\): \((2(4)^2 - 18 = 14) \geq 0\) holds true.

Step 6: Solution to the Inequality

Combine the intervals where the inequality is true. With the critical points included due to \(\geq\), the solution to the inequality is \(x \in (-\infty, -3] \cup [3, \infty)\).

Key Concepts

Solving Inequalities

Inequalities are used to determine where a particular statement or condition holds true for a set of numbers. When dealing with a quadratic inequality like \(2x^2 - 18 \geq 0\), the goal is to find the values of \(x\) that make the inequality true. Think of this as similar to solving a quadratic equation, but instead of finding where expressions are equal, you find where one side is greater than or possibly equal to the other side.

To solve quadratic inequalities, you follow these general steps:

• Identify the quadratic expression involved in the inequality.
• Set the expression equal to zero and solve for the critical points.
• Use these critical points to divide the number line into intervals.
• Test each interval to see if the inequality holds true.
• Combine the intervals where the inequality condition is satisfied.

These steps help to understand where the expression fits conditions set by the inequality and ultimately provides the solution that defines the inequality's valid intervals.

Quadratic Expressions

Quadratic expressions are mathematical phrases that include terms with \(x^2\), representing a parabolic graph. The general form for these expressions is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In this exercise, the quadratic expression is \(2x^2 - 18\).

Key characteristics of quadratic expressions include:

• The degree, which is always 2, because of the squared term \(x^2\).
• The parabola they represent can open upwards or downwards, depending on the sign of \(a\).
• The roots or solutions, which are where the graph crosses the x-axis, if any such intersection exists.

Understanding these characteristics allows you to effectively manipulate and solve quadratic inequalities. For instance, solving \(2x^2 - 18 = 0\) helps find the points that split the expression on the number line, guiding which intervals to explore further.

Inequality Intervals

When solving quadratic inequalities, critical points divide the number line into segments or intervals. In this context, these intervals help us determine where the inequality holds true. After finding that \(x = -3\) and \(x = 3\), these points divide the line into three segments:

• \((-finity, -3)\)
• \([-3, 3]\)
• \((3, \infinity)\)

Choosing test points from within each segment helps us check whether the inequality condition \(2x^2 - 18 \geq 0\) is satisfied for those entire intervals. If a test point in an interval makes the inequality true, the whole interval is a solution. Here, the intervals \((-finity, -3]\) and \([3, \infinity)\) satisfy the inequality as shown by test points like \(-4\) and \(4\), respectively.

These intervals are a crucial part of understanding inequalities because they tell us all possible values for which the inequality is valid. Visualizing this on a number line can greatly help in grasping where solutions exist.