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Chapter 2: Problem 50

Solve the inequality. Then graph the solution set. $$\frac{x^{2}+x-6}{x} \geq 0$$

Short Answer

Expert verified
The solution to the inequality \(\frac{x^{2}+x-6}{x} \geq 0\) is \(x \in (-\infty, -3) \cup (2, \infty)\) and the denominator can't be zero i.e \(x \neq 0\). Draw a number line, indicating these intervals with open circles at -3 and 2 and coloring in the regions to the left of -3 and to the right of 2.

Step by step solution

01

Solve the equation and find critical points

Rearrange the inequality as \(\frac{x^{2}+x-6}{x} - 0 \geq 0\). Equate it to zero to find the critical points. Factor the numerator to rewrite the expression as \(\frac{(x+3)(x-2)}{x} = 0\). Set each factor equal to zero and solve for x: \(x+3=0\) gives \(x=-3\) and \(x=2\) for \(x-2=0\). Also note that \(x \neq 0\) as the denominator cannot be zero.
02

Determine solution intervals and test them

The critical points divide the number line into four intervals: (-鈭�, -3), (-3, 0), (0, 2), and (2, 鈭�). Test the intervals using a test point from each interval and substitute it into the original inequality. Choose -4, -1, 1, and 3 respectively. Only test points from (-鈭�, -3) and (2, 鈭�) satisfy the original inequality.
03

Graph the solution set

Graph the solution set on the number line using open circles for -3 and 2 as the points are not included in the solution set. Color in the regions corresponding to the intervals (-鈭�, -3) and (2, 鈭�). The graph shows the solution set for the given inequality.

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Key Concepts

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

Graphing Inequalities
To understand graphing inequalities, you need to visualize the solution set of an inequality on a number line or coordinate plane. When dealing with inequalities, we're interested in finding all the values of the variable(s) that make the inequality true. For the inequality \(\frac{x^{2}+x-6}{x} \geq 0\), graphing it involves showing the regions where this expression is non-negative.
  • This inequality is expressed as a rational expression, which can be more complex to graph due to its numerator and denominator.
  • The key to graphing is first finding the points where the inequality is exactly zero, known as the critical points.
  • After identifying critical points, the inequality can then be tested within intervals created by these points.
      • By graphing these intervals, you can illustrate which parts of the number line satisfy the inequality, using open or closed circles to denote whether the critical points themselves are included in the solution set.
Critical Points
Critical points are values of \(x\) which make the numerator, denominator, or both, equal to zero in a rational expression. For the inequality \(\frac{(x+3)(x-2)}{x} \geq 0\), there are three critical points:
  • \(x = -3\) and \(x = 2\), where the numerator is zero
  • \(x = 0\), where the denominator is zero
These points are essential as they help to divide the number line into distinct intervals. Critical points are where the inequality may change its truth value. To determine the solution intervals, you test values from each interval to see whether they satisfy the inequality. Remember that if the inequality involves division, points that make the denominator zero cannot be part of the solution, making them open circles on the graph.
Solution Intervals
Solution intervals are sections of the number line that satisfy the inequality. After determining the critical points for \(\frac{(x+3)(x-2)}{x} \geq 0\), the number line is split into four intervals based on these points:
  • \((-\infty, -3)\)
  • \((-3, 0)\)
  • \((0, 2)\)
  • \((2, \infty)\)
For each interval, use a test point to see if it makes the inequality true. In this case, test points like -4 and 3 will show that the intervals \((-\infty, -3)\) and \((2, \infty)\) satisfy \(\frac{(x+3)(x-2)}{x} \geq 0\). These intervals are part of the solution set and should be shaded or noted on the graph, excluding the critical points themselves, which means we use open circles at -3 and 2.

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Most popular questions from this chapter

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