Problem 62 - Algebra for College Students

Question

Solve each inequality. State the solution set using interval notation when possible. \(x^{2} \geq 36\)

Step-by-Step Solution

Answer

The solution set is: \(( -\infty, -6 ] \cup [ 6, \infty )\).

Step 1: Rewrite the Inequality in Standard Form

To start, we have the inequality: \(x^{2} \, \geq \, 36 \).

Step 2: Solve the Inequality As an Equation

Next, solve the corresponding equation: \(x^{2} = 36 \) By taking the square root of both sides, we have: \(x = \, \pm 6 \).

Step 3: Determine the Critical Points

We have found the critical points to be \(x = 6\) and \(x = -6\). These points will divide the number line into three intervals: \((-\infty, -6)\), \([-6, 6]\), and \((6, \infty)\).

Step 4: Test the Intervals

Choose a test point from each interval to determine where the inequality is true. For \((-\infty, -6)\), choose \(x = -7\): \((-7)^2 = 49\). Since \(49 \geq 36\), this interval is part of the solution. For \([-6, 6]\), choose \(x = 0\): \((0)^2 = 0\). Since \(0 ot\geq 36\), this interval is not part of the solution. For \((6, \infty)\), choose \(x = 7\): \((7)^2 = 49\). Since \(49 \geq 36\), this interval is part of the solution.

Step 5: Combine the Solution Intervals

Combine the intervals where the inequality is true. Since the inequality includes \(\geq\), include the endpoints \(x = -6\) and \(x = 6\).

Step 6: Write the Solution Set in Interval Notation

The solution set in interval notation is: \(( -\infty, -6 ] \cup [ 6, \infty )\).

Key Concepts

inequality solutions

Inequality solutions involve finding all values of the variable that make the inequality true. These types of problems are similar to solving equations but add an extra layer of complexity because they deal with ranges of values rather than specific solutions.

• Understanding Inequalities: Inequalities such as \(x^2 \geq 36\) mean we are looking for values of \x\ that make this statement true. This is different from an equation where we look for specific numbers.
• Critical Points: Identifying the critical points helps in dividing the number line into segments. This is crucial for testing where the inequality holds true.
• Testing Intervals: After identifying the critical points, we select test points from each interval. If the inequality holds true for the test point, the entire interval is part of the solution.
• Combining Intervals: Finally, we combine all the intervals where the inequality is true to express the solution set.

By thoroughly understanding these steps, students can solve inequality problems efficiently and correctly.

interval notation

Interval notation is a way of representing a range of values on the number line. It is especially useful for expressing the solution set of inequalities.

• Open and Closed Intervals: An open interval does not include its endpoints (e.g., \(a, b\)), whereas a closed interval includes its endpoints (e.g., \[a, b\]).
• Infinite Intervals: When an interval extends indefinitely, we use the infinity symbol (\

critical points

When solving inequalities, critical points are the values of the variable that satisfy the corresponding equation (converting \geq\, \leq\, >, or < into an equation). These points divide the number line into smaller intervals.

Consider the inequality \(x^2 \geq 36\). Solving its equation version \(x^2 = 36\) gives us the critical points \(x = -6\) and \(x = 6\).

By evaluating these points, we can determine the nature of each interval. Choose a sample point from each interval, plug it back into the inequality, and check if the inequality holds true.

In our case, for \ x^2 \geq 36\, test points in each interval around \ -6\ and \ 6\ give us either \ true \ or \ false \ statements, which decide if the intervals are part of the solution set.

Utilizing critical points effectively allows us to accurately pin down where the inequality is satisfied.