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Chapter 2: Problem 47

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$\frac{-x+2}{x-4} \geq 0$$

Short Answer

Expert verified
The solution to the inequality is \(x \leq 2\), which in interval notation is \((- \infty, 2]\).

Step by step solution

01

Identify Critical Points

The critical points are values of \(x\) which make the inequality equal to zero, or undefined. So, the critical points for this equation are \(x = 2\) (which makes the fraction equal to zero) and \(x = 4\) (which makes the fraction undefined because we cannot divide by zero).
02

Test Intervals

Let's split the number line into three regions by these points, and test a number in each interval. \n Region 1: \(x < 2\), take \(x=1\), Region 2: \(2 < x < 4\), take \(x=3\), Region 3: \(x > 4\), take \(x = 5\)\n Plug these values back into the inequality, if the value makes the fraction positive, then that region is included in the solution. If the value makes the fraction negative, that region is not included in the solution. \n For Region 1, the expression becomes \(-\frac{1-2}{1-4}\) , which is positive. \n For Region 2, the expression becomes \(-\frac{3-2}{3-4}\) , which is negative. \n For Region 3, the expression becomes \(-\frac{5-2}{5-4}\) , which is negative.
03

Write Solution in Interval Notation and Graph

Based on step 2, the only interval where the inequality holds true is when \(x < 2\) and at \(x=2\) since the inequality is greater than or equal to zero. Therefore, in interval notation, the solution is \((- \infty, 2]\). The graph will be a line shaded up to and including the point \(x=2\). The point \(x=4\) could be shown on the graph as a vertical dotted line, because it's not a part of the solution set, it is excluded due it makes the the fraction undefined.

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Most popular questions from this chapter

Use a graphing utility to graph $$f(x)=\frac{x^{2}-4 x+3}{x-2} \text { and } g(x)=\frac{x^{2}-5 x+6}{x-2}$$ What differences do you observe between the graph of \(f\) and the graph of \(g\) ? How do you account for these differences?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \((x+3)^{2}, x \neq-3,\) resulting in the equivalent inequality \((x-2)(x+3)<2(x+3)^{2}\).

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-x+1}{x-1}$$

The rational function $$f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x),\) per 100,000 drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\). a. Graph the function in a [0,70,5] by [0,400,20] viewing rectangle. b. Describe the trend shown by the graph. c. Use the \([\mathrm{ZOOM}]\) and \([\mathrm{TRACE}]\) features or the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per 100,000 drivers, are there for this age group?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.
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