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Chapter 2: Problem 27

Solve the inequality. Then graph the solution set. $$x^{3}-3 x^{2}-x>-3$$

Short Answer

Expert verified
The solution to the inequality \(x^{3}-3 x^{2}-x > -3\) is \(-\infty < x < 1\).

Step by step solution

01

Simplify the Inequality

Rearrange the inequality into the standard format: \(x^3 - 3x^2 - x + 3 > 0\)
02

Solve the Associated Equation

To find the roots of the associated equation, set the left side of the inequality to zero: \(x^3 - 3x^2 - x + 3 = 0\). This polynomial can be factored as \((x-1)^3=0\), and we can see that it has a root, \(x = 1\). This root will serve as a breakpoint to determine the intervals of \(x\).
03

Determine the Sign of Each Interval

We break the number line into intervals using the root. The intervals are: \(-\infty < x < 1\) and \(1 < x < \infty\). We then choose a test point from each interval and substitute it into the inequality. For interval \(-\infty < x < 1\), we can choose \(x = 0\). Then the inequality \(0^3 - 3*0^2 - 0 + 3 > 0\)=> \(3 > 0\) evaluates as true. So the entire interval \(-\infty < x < 1\) satisfies the inequality. For interval \(1 < x < \infty\), choose \(x = 2\). The inequality \(2^3 - 3*2^2 - 2 + 3 > 0\) => \(-1 > 0\) is false. So this interval does not satisfy the inequality.
04

Graph the Solution

Draw a number line and indicate the root with a open circle at \(x = 1\), as the solution does not include the root. Shade the interval \(-\infty < x < 1\), which represents the solution to the inequality.

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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. $$f(x)=x^{3}-x^{2}+x+39$$

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{3}{2} x^{2}-\frac{23}{2} x+6=\frac{1}{2}\left(2 x^{3}-3 x^{2}-23 x+12\right)$$

The coordinate system shown below is called the complex plane. In the complex plane, the point that corresponds to the complex number \(a+b i\) is \((a, b)\) (GRAPH CANNOT COPY) Match each complex number with its corresponding point. (i) 3 (ii) \(3 i\) (iii) \(4+2 i\) (iv) \(2-2 i\) (v) \(-3+3 i\) (vi) \(-1-4 i\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{4}-4 x^{3}+16 x-16\) (a) Upper: \(x=5\) (b) Lower: \(x=-3\)

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{3}+9 x^{2}+27 x+35$$
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