Chapter 1

Solve each equation for \(x\) in terms of the other letters. $$\frac{1}{a x}=\frac{1}{b x}-\frac{1}{c}$$

Express each interval using inequality notation and show the given interval on a number line. $$(-\infty, \infty)$$

Verify the identity $$\left(y_{2}-y_{1}\right) /\left(x_{2}-x_{1}\right)=\left(y_{1}-y_{2}\right) /\left(x_{1}-x_{2}\right)$$ What does this identity tell you about calculating slope?

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-3| \leq 4$$

Find the standard equation of the circle tangent to the \(y\) -axis and with center (3,5)

Solve each equation for \(x\) in terms of the other letters. $$\frac{1}{a}-\frac{1}{x}=\frac{1}{x}-\frac{1}{b}$$

The value of the irrational number \(\pi\), correct to ten decimal places (without rounding off), is \(3.1415926535 .\) By using a calculator, determine to how many decimal places each of the following quantities agrees with \(\pi\) (a) \((4 / 3)^{4}:\) This is the value used for \(\pi\) in the Rhind papyrus, an ancient Babylonian text written about 1650 B.C. (b) \(22 / 7:\) Archimedes \((287-212 \text { B.C. })\) showed that \(223 / 71<\pi<22 / 7 .\) The use of the approximation \(22 / 7\) for \(\pi\) was introduced to the Western world through the writings of Boethius (ca. \(480-520\) ), a Roman philosopher, mathematician, and statesman. Among all fractions with numerators and denominators less than 100 , the fraction \(22 / 7\) is the best approximation to \(\pi\) (c) \(355 / 113:\) This approximation of \(\pi\) was obtained in fifth-century China by Zu Chong-Zhi (430-501) and his son. According to David Wells in The Penguin Dictionary of Curious and Interesting Numbers (Harmondsworth, Middlesex, England: Viking Penguin, Ltd., 1986 ), "This is the best approximation of any fraction below \(103993 / 33102\) " (d) \(\frac{63}{25}\left(\frac{17+15 \sqrt{5}}{7+15 \sqrt{5}}\right):\) This approximation for \(\pi\) was obtained by the Indian mathematician Scrinivasa Ramanujan (1887-1920).

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-1| \leq \frac{1}{2}$$

Find the standard equation of the circle passing through the origin and with center (3,5)

Solve each equation for \(x\) in terms of the other letters. (a) \(y=m x+b,\) where \(m \neq 0\) (b) \(y-y_{1}=m\left(x-x_{1}\right),\) where \(m \neq 0\) (c) \(\frac{x}{a}+\frac{y}{b}=1\) (d) \(A x+B y+C=0,\) where \(A \neq 0\)

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. $$a+b$$

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$\left|x+\frac{1}{3}\right|<\frac{3}{2}$$

Solve each equation for \(x\) in terms of the other letters. $$(a x+b)^{2}-(b x+a)^{2}=0, \text { where } a \neq \pm b$$

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. $$a b$$

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$\left|x+\frac{\pi}{2}\right|>1$$

(a) Verify that the point (3,7) is on the circle $$x^{2}+y^{2}-2 x-6 y-10=0$$ (b) Find the equation of the line tangent to this circle at the point \((3,7) .\) Hint: A result from elementary geometry says that the tangent to a circle is perpendicular to the radius drawn to the point of contact.

Solve each equation for \(x\) in terms of the other letters. $$(x-p)^{2}+(x-q)^{2}=p^{2}+q^{2}$$

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. $$a / b$$

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-5| \geq 2$$

Solve each equation for \(x\) in terms of the other letters. \(a^{2}(a-x)=b^{2}(b+x)-2 a b x,\) where \(a \neq b\)

(a) Give an example in which the result of raising a rational number to a rational power is an irrational number. (b) Give an example in which the result of raising an irrational number to a rational power is a rational number.

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x+5| \geq 2$$

Solve each equation for \(x\) in terms of the other letters. $$\frac{b}{a x-1}-\frac{a}{b x-1}=0, \text { where } a \neq b$$

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. Can an irrational number raised to an irrational power yicld an answer that is rational? This problem shows that the answer is "yes." (However, if you study the following solution very carefully, you'll see that even though we've answered the question in the affirmative, we've not pinpointed the specific case in which an irrational number raised to an irrational power is rational.) (a) Let \(A=(\sqrt{2})^{\sqrt{2}} .\) Now, either \(A\) is rational or \(A\) is irrational. If \(A\) is rational, we are done. Why? (b) If \(A\) is irrational, we are done. Why? Hint: Consider \(A^{\sqrt{2}}\)

In parts (a) and (b), sketch the interval or intervals corresponding to the given inequality: (a) \(|x-2|<1\) (b) \(0<|x-2|<1\) (c) In what way do your answers in (a) and (b) differ? (The distinction is important in the study of limits in calculus.)

Solve each equation for \(x\) in terms of the other letters. $$\frac{a-x}{a-b}-2=\frac{c-x}{b-c}$$

Show that for all real numbers \(a\) and \(b,\) we have $$ |a|-|b| \leq|a-b| $$ Hint: Beginning with the identity \(a=(a-b)+b,\) take the absolute value of each side and then use the triangle inequality.

Solve each equation for \(x\) in terms of the other letters. $$\frac{x+2 p}{2 q-x}+\frac{x-2 p}{2 q+x}-\frac{4 p q}{4 q^{2}-x^{2}}=0$$

Show that $$ |a+b+c| \leq|a|+|b|+|c| $$ for all real numbers \(a, b,\) and \(c .\) Hint: The left-hand side can be written \(|a+(b+c)| .\) Now use the triangle inequality.

Solve each equation for \(x\) in terms of the other letters. $$\frac{x-a}{x-b}=\frac{b-x}{a-x}, \text { where } a \neq b$$

Explain why there are no real numbers that satisfy the equation \(\left|x^{2}+4 x\right|=-12\).

Solve each equation for \(x\) in terms of the other letters. $$1-\frac{a}{b}\left(1-\frac{a}{x}\right)-\frac{b}{a}\left(1-\frac{b}{x}\right)=0$$

(As background for this exercise, you might want to work Exercise \(23 .)\) Prove that $$ \max (a, b)=\frac{a+b+|a-b|}{2} $$ Hint: Consider three separate cases: \(a=b ; a>b ;\) and \(b>a\).

Solve each equation for the indicated variable. \(S=2 \pi r^{2}+2 \pi r h ;\) for \(h\)

(As background for this exercise, you might want to work Exercise \(24 .\) ) Prove that $$ \min (a, b)=\frac{a+b-|a-b|}{2} $$.

Solve each equation for the indicated variable. $$\frac{x_{1} x}{a^{2}}+\frac{y_{1} y}{b^{2}}=1 ; \text { for } y$$

Solve each equation for the indicated variable. $$d=\frac{r}{1+r t} ; \text { for } r$$

Solve each equation for the indicated variable. $$S=\frac{r l-a}{r-1} ; \text { for } r$$

Solve the equations (In these exercises, you'll need to multiply both sides of the equations by expressions involving the variable. Remember to check your answers in these cases.) $$\frac{3}{x+5}+\frac{4}{x}=2$$

Solve the equations (In these exercises, you'll need to multiply both sides of the equations by expressions involving the variable. Remember to check your answers in these cases.) $$\frac{5}{x+2}-\frac{2 x-1}{5}=0$$

Solve the equations (In these exercises, you'll need to multiply both sides of the equations by expressions involving the variable. Remember to check your answers in these cases.) $$1-x-\frac{2}{6 x+1}=0$$

Solve the equations (In these exercises, you'll need to multiply both sides of the equations by expressions involving the variable. Remember to check your answers in these cases.) $$\frac{x^{2}-3 x}{x+1}=\frac{4}{x+1}$$

Solve the equations (In these exercises, you'll need to multiply both sides of the equations by expressions involving the variable. Remember to check your answers in these cases.) $$\frac{x}{x-2}+\frac{x}{x+2}=\frac{8}{x^{2}-4}$$

Solve the equations (In these exercises, you'll need to multiply both sides of the equations by expressions involving the variable. Remember to check your answers in these cases.) $$\frac{2 x}{x^{2}-1}-\frac{1}{x+3}=0$$

Given the equation \(\frac{1}{x}=\frac{1}{a}+\frac{1}{b}\) (a) Solve to show \(x=\frac{a b}{a+b},\) provided \(a+b \neq 0\) (b) Check the solution.