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Chapter 2: Problem 25

Solving a Polynomial Inequality In Exercises \(13-34,\) solve the inequality. Then graph the solution set.$$x^{2}-3 x-18>0$$

Short Answer

Expert verified
The solution to the inequality \(x^{2}-3x-18>0\) is \( x<-3 \) or \( x>6 \)

Step by step solution

01

Find the roots of the inequality

The roots of the inequality \(x^{2}-3x-18=0\) can be found using the quadratic formula: \(x=\frac{-(-3)\pm \sqrt{(-3)^{2}-4(1)(-18)}}{2(1)}\). Solving gives \(x=6\) and \(x=-3\).
02

Divide the number line into intervals

The roots divide the number line into three intervals. The intervals are \((- \infty,-3)\), \((-3,6)\) and \((6,\infty)\).
03

Test the intervals

Choose a test point from each interval and substitute it into the original inequality to find out whether the inequality holds. For \((- \infty,-3)\), take \(x=-4\) for example, plug in the values and see if the inequality stands. Repeat this step for \((-3,6)\), for example choose \(x=0\), and for \((6,\infty)\), for example, choose \(x=7\).
04

Interpret the results and draw the graph

The intervals that make the inequality true form the solution set. Draw a number line and shade the solution set. Open circle is used at the points -3 and 6 because original inequality is not inclusive i.e no 'or equal to' given.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of a quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\). In this exercise, we use it to solve the equation \(x^2 - 3x - 18 = 0\). The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] When using the formula:
  • \(a\) is the coefficient of \(x^2\), which is 1 in our equation.
  • \(b\) is the coefficient of \(x\), which is -3.
  • \(c\) is the constant term, which is -18.
By substituting these values into the formula, we compute:\[x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-18)}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 + 72}}{2}\]Which results in the roots \(x = 6\) and \(x = -3\). These are the points where the parabola crosses the x-axis.
Inequality Intervals
Solving polynomial inequalities involves determining where the polynomial is greater than, less than, or equal to zero in certain intervals. For \(x^2 - 3x - 18 > 0\), the roots \(x = 6\) and \(x = -3\) help to divide the number line into distinct intervals.These intervals are:
  • \((-\infty, -3)\)
  • \((-3, 6)\)
  • \((6, \infty)\)
Each interval represents a portion of the number line where the original inequality might hold true. Finding which intervals satisfy the inequality is the next step.
Number Line Graphing
Graphing inequalities helps visualize where the inequality is true on a number line. After determining the intervals \((-\infty, -3)\), \((-3, 6)\), and \((6, \infty)\), we choose test points from each interval and substitute them into the original inequality \(x^2 - 3x - 18 > 0\).For instance:
  • In \((-\infty, -3)\), pick \(x = -4\): Substitute to find if \((-4)^2 - 3(-4) - 18 > 0\).
  • In \((-3, 6)\), pick \(x = 0\): Test to see if \(0^2 - 3(0) - 18 > 0\).
  • In \((6, \infty)\), choose \(x = 7\): Calculate if \(7^2 - 3(7) - 18 > 0\).
Each calculation confirms the solution set for the inequality. Graphically, shaded portions on the number line indicate where the inequality is satisfied.
Solution Set Interpretation
Once the test points have been evaluated, those intervals where the inequality holds make up the solution set. In this case, the inequality \(x^2 - 3x - 18 > 0\) is satisfied in the intervals \((-\infty, -3)\) and \((6, \infty)\).When graphing:
  • Use open circles (not filled in) at the test points \(-3\) and \(6\) since the inequality is strict (\(>\), not \(\geq\)).
  • Shade the regions outside these points on the number line.
The visual representation on the number line allows us to clearly see that the polynomial is positive in these intervals. Understanding this interpretation solidifies the concept of solving polynomial inequalities through graphical analysis.

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