Problem 17 - Beginning and Intermediate Algebra

Question

Solve each quadratic inequality. Graph the solution set and write the solution in interval notation. $$b^{2}-9 b>0$$

Step-by-Step Solution

Answer

The solution to the quadratic inequality \(b^2 - 9b > 0\) is \((- \infty, 0) \cup (9, \infty)\). The graph has open circles at \(b = 0\) and \(b = 9\), with shading to the left of \(b = 0\) and to the right of \(b = 9\).

Step 1: Find the critical points

First, set the inequality as an equation and solve for the critical points: $$b^2 - 9b = 0$$ This equation can be factored as: $$b(b - 9) = 0$$ Which gives us the critical points: $$b = 0, b = 9$$

Step 2: Test intervals

Divide the number line into intervals based on the critical points \(b = 0\) and \(b = 9\). Test a value from each interval in the original inequality to see if it holds true. - Interval 1: \(b < 0\) Test with \(b = -1\). Does \((-1)^2 - 9(-1) > 0\) hold true? \(1 + 9 = 10 > 0\), so this interval satisfies the inequality. - Interval 2: \(0 < b < 9\) Test with \(b = 1\). Does \((1)^2 - 9(1) > 0\) hold true? \(1 - 9 = -8 < 0\), so this interval does not satisfy the inequality. - Interval 3: \(b > 9\) Test with \(b = 10\). Does \((10)^2 - 9(10) > 0\) hold true? \(100 - 90 = 10 > 0\), so this interval satisfies the inequality.

Step 3: Graph the solution set

Since the inequality is greater than zero (rather than greater or equal), we represent the critical points with open circles on the number line. We shade the intervals that satisfy the inequality (Intervals 1 and 3): - An open circle at \(b = 0\) - Shade to the left of \(b = 0\) - An open circle at \(b = 9\) - Shade to the right of \(b = 9\)

Step 4: Write the solution in interval notation

The solution in interval notation is the union of the two intervals that satisfy the inequality: \((- \infty, 0) \cup (9, \infty)\)

Key Concepts

Interval Notation

Interval notation is a mathematical way of representing a set of numbers along the real number line. It is highly efficient for showcasing solution sets, especially those that involve inequalities. In interval notation, we use parentheses

• \((a, b)\) to denote an interval excluding the endpoints \(a\) and \(b\), meaning the values are greater than \(a\) and less than \(b\).
• \([a, b]\) to include both endpoints \(a\) and \(b\), translating to values greater than or equal to \(a\) and less than or equal to \(b\).
• Using \((a, \infty)\) or \((-\infty, b)\) implies that the interval stretches indefinitely in a positive or negative direction without including the endpoint \(b\) or negative infinity.

Breaks like these in intervals explain the behavior and solutions of inequalities. They offer a coherent, straightforward view of solutions without the need for exhaustive descriptions.

Solution Set

The solution set of an inequality is the collection of all values that satisfy the inequality. In dealing with quadratic inequalities like

• \(b^2 - 9b > 0\),

it's essential to first find the critical points by treating the inequality as an equation. Here, the critical points were \(b = 0\) and \(b = 9\). These values serve as pivotal markers directing us to segment the number line into intervals. By assessing each interval, we determine where the inequality holds true. Values from the intervals

• \((-\infty, 0)\)
• \((9, \infty)\)

fulfilled the condition. This means every number in these intervals makes the inequality positive. Consequently, our solution set encompasses both intervals, demonstrating the regions where the quadratic expression is greater than zero.

Test Intervals

Testing intervals is a systematic technique used in solving inequalities. After identifying the critical points from an equation like

• \(b^2 - 9b = 0\),

we divide the number line into sections defined by these points:

• \(b < 0\),
• \(0 < b < 9\),
• \(b > 9\)

For each interval, we choose a test value and substitute it back into the original inequality:

• For \(b < 0\): Test with \(b = -1\), results in \(1 + 9 = 10\) (true).
• For \(0 < b < 9\): Test with \(b = 1\), results in \(-8\) (false).
• For \(b > 9\): Test with \(b = 10\), results in \(10\) (true).

This process confirms which intervals satisfy the original inequality, key for determining the solution set. It's a strategic way to analyze and confirm plausible solutions.

Graphing Inequalities

Graphing inequalities offers a visual representation of the solution set. This involves plotting regions of the number line that satisfy the inequality, using open or closed circles. Open circles indicate that the endpoint is not included in the solution, typical for '<', '>', symbols, as seen with

• \(b = 0\) and \(b = 9\) in \(b^2 - 9b > 0\).

After identifying these points, shading helps convey which sections satisfy the inequality. For this example, shaded regions were:

• left of \(b = 0\) (i.e., \((-\infty, 0)\)),
• right of \(b = 9\) (i.e., \((9, \infty)\)).

This graphing method offers a clear, intuitive way to visualize solutions and confirm the results of algebraic findings. By examining the graph, one can easily verify the accuracy of their computations and gain a deeper understanding of the nature of the inequality's solutions.