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Chapter 2: Problem 43

Solve the inequality. Then graph the solution set. $$\frac{x+6}{x+1}-2<0$$

Short Answer

Expert verified
The solution to the inequality is \(x > 4\). This is graphed by a line shaded towards the positive infinity from a solid point at \(x = 4\).

Step by step solution

01

Simplify the inequality

Begin by simplifying the given inequality. This can be achieved by subtracting -2 from both sides of the inequality, which leads to \(\frac{x+6}{x+1}-2+2<0+2\). This simplifies to \(\frac{x+6}{x+1}<2\).
02

Clear the fraction

Clear the fraction in the simplified inequality by multiplying everything by the denominator, yielding: \((x+6)<2(x+1)\) which simplifies to \(x + 6 < 2x + 2\).
03

Solve for x

Solving for x involves subtracting x from both sides, which results in \((x+6)-x<2x+2-x\) that simplifies to \(6 < x + 2\). Subtracting 2 from both sides results in \(6-2 4\). During these manipulations, be careful to reverse inequality direction each time you multiply or divide by a negative.
04

Identify critical numbers and intervals

The critical numbers include where the inequality equals to zero and where it's undefined. The inequality is zero when the numerator is \(x = -6\) and undefined when the denominator equals to zero given by \(x = -1\). Notice that these critical points divided the number line into three intervals: \(-\infty \leq x < -1\), \(-1 < x < 4\), and \(4 < x \leq +\infty\).
05

Check each interval

Test a random number from each interval in the original inequality. Use logical 'or' if the inequality is less than or greater than; use logical 'and' if it is less than or equal to, or greater than or equal to. If we pick \(-2\), \(0\), and \(5\) respectively; Only \(5\) satisfies the inequality. So the valid interval is \(4 < x \leq +\infty\).
06

Graph the solution set

The graph should be plotted on a number line. The number line should have a solid circle at \(x = 4\) and be shaded to the right towards \(\infty\).

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}-16$$

Sketch the graph of each polynomial function. Then count the number of real zeros of the function and the numbers of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? (a) \(f(x)=-x^{3}+9 x\) (b) \(f(x)=x^{4}-10 x^{2}+9\) (c) \(f(x)=x^{5}-16 x\)

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}+10 x^{2}+9$$

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(\left.x^{2}+4 .\right)\)

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$h(x)=2 x^{4}-3 x+2$$
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