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Chapter 2: Problem 31

Solve the inequality. Then graph the solution set. $$x^{3}-4 x \geq 0$$

Short Answer

Expert verified
The solution to the inequality \(x^3 - 4x \geq 0\) is \(-2 \leq x \leq 0\) and \(x \geq 2\)

Step by step solution

01

Find the roots

The cubic function can be factored into \(x(x-2)(x+2)\). Setting each factor equal to zero gives us the roots \(x=0\), \(x=2\), and \(x=-2\).
02

Analyse the sign changes

We plot these roots on a number line. This divides the number line into four intervals: \(-\infty < x < -2\), \(-2 < x < 0\), \(0 < x < 2\), and \(2 < x < \infty\). We pick a test value in each interval and substitute it into the inequality to analyse how the function changes its sign.
03

Make a sign table

For the interval \(-\infty < x < -2\) we can pick -3, when substituted into the original inequality, x^3 - 4x is negative. Hence, all values in this interval are negative. Similarly, for the other intervals, we find that \(-2 < x < 0 \), \(x\) is positive, for \(0 < x < 2\), \(x\) is negative and for \(2 < x < \infty\), \(x\) is positive.
04

Find the solution set

As we are solving for \(x^3 - 4x \geq 0\), the solution set would include values of \(x\) where the function is positive and the values where function equals zero, i.e. the roots. Hence, the solution to the inequality are all \(x\) such that \(-2 \leq x \leq 0\) and \(x \geq 2\).
05

Graph the solution

The final step involves drawing a number line, marking the solution set obtained above and filling in those sections. The roots at x = -2, x = 0 and x = 2 are marked with a solid dot as they are included in the solution set, and the line is drawn towards +infinity from 2 and between -2 and 0.

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Most popular questions from this chapter

Explore transformations of the form \(g(x)=a(x-h)^{5}+k\) (a) Use a graphing utility to graph the functions \(y_{1}=-\frac{1}{3}(x-2)^{5}+1\) and \(y_{2}=\frac{3}{5}(x+2)^{5}-3\) Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of \(g\) always be increasing or decreasing? If so, then is this behavior determined by \(a, h,\) or \(k ?\) Explain. (c) Use the graphing utility to graph the function \(H(x)=x^{5}-3 x^{3}+2 x+1\) Use the graph and the result of part (b) to determine whether \(H\) can be written in the form \(H(x)=a(x-h)^{5}+k\) Explain.

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{2}+36$$

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

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x+4=0$$

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=-f(x)$$
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