Chapter 1

Write each statement as an absolute value equation or inequality. \(q\) is no more than 8 units from 22.

Solve each rational inequality. Write each solution set in interval notation.4 $$\frac{9 x-8}{4 x^{2}+25}<0$4

Solve each equation. $$7 x^{-2}-10 x^{-1}-8=0$$

Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real complex. Do not solve the equation. $$2 x^{2}+4 x+1=0$$

Simplify each power of i. $$i^{23}$$

Suppose that \(y=5 x+1\) and we want \(y\) to be within 0.002 unit of \(6 .\) For what values of \(x\) will this be true?

Solve each equation. $$x^{-2 / 3}+x^{-1 / 3}-6=0$$

Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real complex. Do not solve the equation. $$9 x^{2}+11 x+4=0$$

Answer each of the following Suppose \(f(x)\) is the number of cars that can be built for \(x\) dollars.What does \(f^{-1}(1000)\) represent?

Simplify each power of i. $$i^{27}$$

Solve each rational inequality. Write each solution set in interval notation.4 $$\frac{(5-3 x)^{2}}{(2 x-5)^{3}}>0$$

Solve each equation. $$2 x^{-2 / 5}-x^{-1 / 5}-1=0$$

Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real complex. Do not solve the equation. $$3 x^{2}=4 x-5$$

Answer each of the following Suppose \(f(r)\) is the volume (in cubic inches) of a sphere of radius \(r\) inches. What does \(f^{-1}(5)\) represent?

Simplify each power of i. $$i^{32}$$

Solve each problem. Dr. Tydings has found that, over the years, \(95 \%\) of the babies he has delivered weighed \(x\) pounds, where $$|x-8.2| \leq 1.5.$$ What range of weights corresponds to this inequality?

Solve each rational inequality. Write each solution set in interval notation.4 $$9\frac{(5 x-3)^{3}}{(25-8 x)^{2}} \leq 0$$ $$\frac{(2 x-3)(3 x+8)}{(x-6)^{3}} \geq 0 \quad\( 92. \)\frac{(9 x-11)(2 x+7)}{(3 x-8)^{3}}>0$$.

Solve each equation. $$16 x^{-4}-65 x^{-2}+4=0$$

Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real complex. Do not solve the equation. $$8 x^{2}-72=0$$

Simplify each power of i. $$i^{-13}$$

Solve each problem. The temperatures on the surface of Mars in degrees Celsius approximately satisfy the inequality \(|C+84| \leq 56 .\) What range of temperatures corresponds to this inequality?

Solve each rational inequality. Write each solution set in interval notation.4 $$\frac{(9 x-11)(2 x+7)}{(3 x-8)^{3}}>0$$

Solve each equation. $$625 x^{-4}-125 x^{-2}+4=0$$

Show that the discriminant for the equation $$ \sqrt{2} x^{2}+5 x-3 \sqrt{2}=0 $$ is 49. If this equation is completely solved, it can be shown that the solution set is \(\left\\{-3 \sqrt{2}, \frac{\sqrt{2}}{2}\right\\} .\) We have a discriminant that is positive and a perfect square, yet the two solutions are irrational. Does this contradict the discussion in this section? Explain.

Answer each of the following For a one-to-one function \(f,\) find \(\left(f^{-1} \circ f\right)(2),\) where \(f(2)=3\)

Simplify each power of i. $$i^{-13}$$

Solve each problem. The industrial process that is used to convert methanol to gasoline is carried out at a temperature range of \(680^{\circ} \mathrm{F}\) to \(780^{\circ} \mathrm{F}\). Using \(F\) as the variable, write an absolute value inequality that corresponds to this range.

In this section we introduced methods of solving equations quadratic in form by substitution and solving equations involving radicals by raising each side of the equation to a power. Suppose we wish to solve $$x-\sqrt{x}-12=0$$ We can solve this equation using either of the two methods. Work Exercises \(93-96\) in order to see how both methods apply. Let \(u=\sqrt{x}\) and solve the equation by substitution. What is the value of \(u\) that does not lead to a solution of the equation?

Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse. $$\begin{array}{l} f(x)=6 x^{3}+11 x^{2}-6 \\ {[-3,2] \text { by }[-10,10]} \end{array}$$

Simplify each power of i. $$i^{-14}$$

Solve each problem. When a model kite was flown in crosswinds in tests to determine its limits of power extraction, it attained speeds of 98 to \(148 \mathrm{ft}\) per sec in winds of 16 to \(26 \mathrm{ft}\) per sec. Using \(x\) as the variable in each case, write absolute value inequalities that correspond to these ranges.

Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse. $$\begin{array}{l} f(x)=x^{4}-5 x^{2} \\ {[-3,3] \text { by }[-8,8]} \end{array}$$

Simplify each power of i. $$\frac{1}{i^{-11}}$$

When humans breathe, carbon dioxide is emitted. In one study, the emission rates of carbon dioxide by college students were measured during both lectures and exams. The average individual rate \(R_{L}\) (in grams per hour) during a lecture class satisfied the inequality $$\left|R_{L}-26.75\right| \leq 1.42,$$ whereas during an exam the rate \(R_{E}\) satisfied the inequality $$\left|R_{E}-38.75\right| \leq 2.17.$$ Use this information in Exercises. Find the range of values for \(R_{L}\) and \(R_{E}\).

A projectile is fired straight up from ground level. After \(t\) seconds, its height above the ground is \(s\) feet, where $$s=-16 t^{2}+220 t$$ For what time period is the projectile at least \(624 \mathrm{ft}\) above the ground?

Find the values of \(a, b,\) and \(c\) for which the quadratic equation $$ a x^{2}+b x+c=0 $$ has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.) $$4,5$$

Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse. $$\begin{array}{l} f(x)=\frac{x-5}{x+3}, \quad x \neq-3 \\ {[-8,8] \text { by }[-6,8]} \end{array}$$

Simplify each power of i. $$\frac{1}{i^{-12}}$$

When humans breathe, carbon dioxide is emitted. In one study, the emission rates of carbon dioxide by college students were measured during both lectures and exams. The average individual rate \(R_{L}\) (in grams per hour) during a lecture class satisfied the inequality $$\left|R_{L}-26.75\right| \leq 1.42,$$ whereas during an exam the rate \(R_{E}\) satisfied the inequality $$\left|R_{E}-38.75\right| \leq 2.17.$$ Use this information in Exercises. The class had 225 students. If \(T_{L}\) and \(T_{E}\) represent the total amounts of carbon dioxide in grams emitted during a one-hour lecture and exam, respectively, write inequalities that model the ranges for \(T_{L}\) and \(T_{E}\).

Find the values of \(a, b,\) and \(c\) for which the quadratic equation $$ a x^{2}+b x+c=0 $$ has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.) $$-3,2$$

Velocity of an Object Suppose the velocity, \(v,\) of an object is given by $$v=2 t^{2}-5 t-12$$ vhere \(t\) is time in seconds. (Here \(t\) can be positive or negative.) Find the intervwhere the velocity is negative.

Solve each equation for the indicated variable. Assume all denominators are nonzero. $$d=k \sqrt{h}, \quad \text { for } h$$

Find the values of \(a, b,\) and \(c\) for which the quadratic equation $$ a x^{2}+b x+c=0 $$ has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.) $$1+\sqrt{2}, 1-\sqrt{2}$$

Explain why the following method of simplifying \(i^{-42}\) works. $$ i^{-42}=\frac{1}{i^{42}}=\frac{1}{\left(i^{2}\right)^{21}}=\frac{1}{(-1)^{21}}=\frac{1}{-1}=-1 $$

Velocity of an Object The velocity of an object, \(v,\) after \(t\) seconds is given by $$ v=3 t^{2}-18 t+24 $$ Find the interval where the velocity is negative.

Solve each equation for the indicated variable. Assume all denominators are nonzero. $$x^{2 / 3}+y^{2 / 3}=a^{2 / 3}, \quad \text { for } y$$

Find the values of \(a, b,\) and \(c\) for which the quadratic equation $$ a x^{2}+b x+c=0 $$ has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.) $$i,-i$$

Solve each equation for the indicated variable. Assume all denominators are nonzero. $$\frac{1}{R}=\frac{1}{r_{1}}+\frac{1}{r_{2}}, \quad \text { for } R$$

Show that \(-2+i\) is a solution of the equation \(x^{2}+4 x+5=0\)

Solve each equation for the indicated variable. Assume all denominators are nonzero. $$\frac{E}{e}=\frac{R+r}{r}, \quad \text { for } e$$