Chapter 3

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. $$(a) 2 x^{2}-9 x>-4$$ $$(b) $2 x^{2}-9 x \leq-4$$

Explain why the method of dividing complex numbers (that is, multiplying both the numerator and the denominator by the conjugate of the denominator) works. What property justifies this process?

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. $$(a) 3 x^{2}+13 x+10 \leq 0$$ $$(b) 3 x^{2}+13 x+10>0$$

Suppose that your friend describes a method of simplifying a positive power of \(i\). "Just divide the exponent by 4 , and then look at the remainder. Then, refer to the short table of powers of \(i\) in this section. The given power of \(i\) is equal to \(i\) to the power indicated by the remainder. And if the remainder is \(0,\) the result is \(i^{0}=1 .^{\prime \prime}\) Explain why this method works.

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. $$(a) -x^{2}-x \leq 0$$ $$(b) -x^{2}-x>0$$

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(-x^{2}+2 x \leq 0\) (b) \(-x^{2}+2 x>0\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}-x+1<0\) (b) \(x^{2}-x+1 \geq 0\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x^{2}-x+3<0\) (b) \(2 x^{2}-x+3 \geq 0\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x+1 \geq x^{2}\) (b) \(2 x+1

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}+5 x<2\) (b) \(x^{2}+5 x \geq 2\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x-3 x^{2}>-1\) (b) \(x-3 x^{2} \leq-1\)

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(-2 x^{2}+3 x<-4\) (b) \(-2 x^{2}+3 x \geq-4\)

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$s=\frac{1}{2} g t^{2} \quad \text { for } t$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$\mathscr{A}=\pi r^{2} \quad \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$a^{2}+b^{2}=c^{2} \quad \text { for } a$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$\mathscr{A}=s^{2} \text { for } s$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$S=4 \pi r^{2} \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$V=\frac{1}{3} \pi r^{2} h \quad \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$V=e^{3} \quad \text { for } e$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$V=\frac{4}{3} \pi r^{3} \quad \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$F=\frac{k M v^{4}}{r} \text { for } v$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$s=s_{0}+g t^{2}+k \quad \text { for } t$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$P=\frac{E^{2} R}{(r+R)^{2}} \text { for } R$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$S=2 \pi r h+2 \pi r^{2} \quad \text { for } r$$

Solve each equation for \(x\) and then for \(y\). $$x^{2}+x y+y^{2}=0 \quad(x>0, y>0)$$

Solve each problem. Air Density As the altitude increases, air becomes thinner, or less dense. An approximation of the density \(d\) of air at an altitude of \(x\) meters above sea level is $$d(x)=\left(3.32 \times 10^{-9}\right) x^{2}-\left(1.14 \times 10^{-4}\right) x+1.22$$ The output is the density of air in kilograms per cubic meter. The domain of \(d\) is \(0 \leq x \leq 10,000 .\) (Source: A. Miller and J. Thompson, Elements of Meteorology.) (a) Denver is sometimes referred to as the mile-high city. Compare the density of air at sea level and in Denver. (Hint: \(1 \mathrm{ft} \approx 0.305 \mathrm{m}\) ) (b) Determine the altitudes where the density is greater than 1 kilogram per cubic meter.

Solve each problem. Suppose that a person's heart rate, \(x\) minutes after vigorous exercise has stopped, can be modeled by $$f(x)=\frac{4}{5}(x-10)^{2}+80$$ The output is in beats per minute, where the domain of \(f\) is \(0 \leq x \leq 10\) (a) Evaluate \(f(0)\) and \(f(2) .\) Interpret the result. (b) Estimate the times when the person's heart rate was between 100 and 120 beats per minute, inclusive.

Solve each problem. The table shows a person's heart rate during the first 4 minutes after exercise has stopped. $$\begin{array}{|l|c|c|c|}\hline \text { Time (min) } & 0 & 2 & 4 \\\\\hline \text { Heart rate (bpm) } & 154 & 106 & 90\end{array}$$ (a) Find a formula \(f(x)=a(x-h)^{2}+k\) that models the data, where \(x\) represents time and \(0 \leq x \leq 4\) (b) Evaluate \(f(1)\) and interpret the result. (c) Estimate the times when the heart rate was from 115 to 125 beats per minute.