Problem 21 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ x^{2}-3 x-18>0 $$

Step-by-Step Solution

Answer

\((-\infty, -3) \cup (6, \infty)\)

Step 1: Factor the Quadratic Expression

To solve the inequality \( x^2 - 3x - 18 > 0 \), we'll first factor the quadratic expression. The expression can be factored as \( (x - 6)(x + 3) \) by finding two numbers that multiply to -18 and add to -3.

Step 2: Find the Critical Points

Set each factor equal to zero to find the critical points: \( x - 6 = 0 \) gives \( x = 6 \), and \( x + 3 = 0 \) gives \( x = -3 \). These are the critical points where the sign of the expression can change.

Step 3: Test the Intervals

The critical points divide the number line into three intervals: \( (-\infty, -3) \), \( (-3, 6) \), and \( (6, \infty) \). Test one value from each interval in the inequality \( (x - 6)(x + 3) > 0 \). For \( x = -4 \), the expression is positive; for \( x = 0 \), it is negative; for \( x = 7 \), it is positive.

Step 4: Interpret the Solution

Since the inequality is \( > 0 \), we include intervals where the expression is positive. These are \( (-\infty, -3) \cup (6, \infty) \) while excluding the critical points where the expression equals zero.

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting the expression as a product of two smaller binomials. This makes solving the inequality easier. For the expression \( x^2 - 3x - 18 \), we need to find two numbers that multiply to \(-18\) and add up to \(-3\). In this case, \(-6\) and \(+3\) fit the bill. Therefore, we can factor the expression as \((x - 6)(x + 3)\). Factoring simplifies complex quadratic expressions, allowing us to break them into simpler components. This step is crucial in making the problem easier to manage and is often the first move in algebraically solving quadratic inequalities. Remember, the factors found will give crucial information about the roots or critical points of the expression.

Finding Critical Points

To find critical points, set each factor of the quadratic equation to zero. These points are key because they are the x-values where the expression changes its sign. For the factored expression \((x - 6)(x + 3)\), set \(x - 6 = 0\) which results in \(x = 6\), and set \(x + 3 = 0\) which results in \(x = -3\). These values are the critical points.Critical points are important because they provide boundaries on the number line where we need to test the sign of the expression. They help break the problem into manageable intervals. In essence, these points allow you to understand when the expression transitions from positive to negative and vice versa.

Testing Intervals

After determining the critical points, divide the number line into intervals and test each within the inequality. The critical points \(-3\) and \(6\) divide the line into three intervals: \((-\infty, -3)\), \((-3, 6)\), and \((6, \infty)\).Pick any number in each interval, substitute it into the factored inequality \((x - 6)(x + 3) > 0\), and test its truth:

• Choose \(x = -4\) in \((-\infty, -3)\): The expression is positive.
• Choose \(x = 0\) in \((-3, 6)\): The expression is negative.
• Choose \(x = 7\) in \((6, \infty)\): The expression is positive.

Testing intervals help identify where the quadratic expression satisfies the inequality \(> 0\). Only intervals where the results are positive are solutions to our inequality.

Solving Inequalities Algebraically

Once you test all intervals, interpret the results to solve the inequality algebraically. We determined the intervals \((-\infty, -3)\) and \((6, \infty)\) had positive results. Since the inequality is \(> 0\), we are interested in those positive intervals and can thus exclude the critical values where the expression equals zero.Conclusively, the solution for the inequality \(x^2 - 3x - 18 > 0\) is \((-\infty, -3) \cup (6, \infty)\). Make sure to exclude the critical points in this situation, as they equate the expression to zero but do not satisfy the strict inequality. Solving inequalities algebraically this way gives a methodical approach to finding solution sets without graphing, ensuring accuracy and simplicity.