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Chapter 3: Problem 90

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ \frac{x+2}{x-3} \leq 2 $$

Short Answer

Expert verified
The solutions to the inequality are \(x \in (-\infty , 3)\) and \(x \in (4, \infty)\)

Step by step solution

01

Arrange the inequality

First, the inequality needs to be in a form that can be easily graphed. So arrange the original inequality \(\frac{x+2}{x-3} \leq 2\) to the zero form by subtracting 2 from both sides: \(\frac{x+2}{x-3} - 2 \leq 0\). This can be further simplified to \(\frac{x+2 -2x +6}{x-3} \leq 0\), or \(\frac{4-x}{x-3} \leq 0\).
02

Graph the inequality

Now, graph the function \(f(x)= \frac{4-x}{x-3}\) using a graphing tool. Pay close attention to the regions where \(f(x)\) is either above or below the x-axis. This will help identify the intervals that satisfy the inequality. Remember, within the graph, the function is less than or equal to zero where it is on or below the x-axis, and is greater than zero where it is above the x-axis.
03

Identify the root

Next, find the roots of the inequality, these are the points where the function crosses the x-axis. The roots can also be found algebraically by setting \(f(x) = 0\) and solving for \(x\), in this case \(x = 4\). It's also important to notice where the function is undefined, this is when the denominator is zero. In this case, it is \(x = 3\). So we have two points to consider \((3, 4)\).
04

Analyze the intervals

Using the x-values found in step 3, divide the number line into intervals. Test a number in each interval by substituting it into the inequality to see if it results in a true statement. Thereby, the graph shows that the function is below zero on two intervals: \((-\infty , 3)\) and \((4, \infty)\). Thus these are the solutions to the inequality.

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Most popular questions from this chapter

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=-x^{2}(x+2)(x-2)$$

In Exercises 41鈥�64, a. Use the Leading Coefficient Test to determine the graph鈥檚 end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=x^{2}(x-1)^{3}(x+2)$$

Can the graph of a polynomial function have no y@intercept? Explain.

Exercises 113鈥�115 will help you prepare for the material covered in the next section. Divide 737 by 21 without using a calculator. Write the answer as quotient \(+\frac{\text { remainder }}{\text { divisor }}\)

Exercises 113鈥�115 will help you prepare for the material covered in the next section. Rewrite \(4-5 x-x^{2}+6 x^{3}\) in descending powers of \(x\)
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