Chapter 1

Find the solution sets of the given inequalities. $$ \left|\frac{2 x}{7}-5\right| \geq 7 $$

Change each repeating decimal to a ratio of two integers \(2.56565656 \ldots\)

Find the value of \(c\) for which the line \(3 x+c y=5\) (a) passes through the point \((3,1)\); (b) is parallel to the \(y\)-axis; (c) is parallel to the line \(2 x+y=-1\); (d) has equal \(x\) - and \(y\)-intercepts; (e) is perpendicular to the line \(y-2=3(x+3)\).

Find the distance between the points on the circle \(x^{2}+y^{2}=13\) with the \(x\)-coordinates \(-2\) and 2 . How many such distances are there?

Find the solution sets of the given inequalities. $$ \left|\frac{x}{4}+1\right|<1 $$

Change each repeating decimal to a ratio of two integers \(3.929292 \ldots\)

Write the equation for the line through \((-2,-1)\) that is perpendicular to the line \(y+3=-\frac{2}{3}(x-5)\).

Find the distance between the points on the circle \(x^{2}+2 x+y^{2}-2 y=20\) with the \(x\)-coordinates \(-2\) and 2 . How many such distances are there?

Find the solution sets of the given inequalities. $$ |5 x-6|>1 $$

Change each repeating decimal to a ratio of two integers \(0.199999 \ldots\)

Find the value of \(k\) such that the line \(k x-3 y=10\) (a) is parallel to the line \(y=2 x+4\); (b) is perpendicular to the line \(y=2 x+4\); (c) is perpendicular to the line \(2 x+3 y=6\).

Which of the following functions satisfies \(f(x+y)=f(x)+f(y)\) for all real numbers \(x\) and \(y ?\) (a) \(f(t)=2 t\) (b) \(f(t)=t^{2}\) (c) \(f(t)=2 t+1\) (d) \(f(t)=-3 t\)

Find the solution sets of the given inequalities. $$ |2 x-7|>3 $$

Change each repeating decimal to a ratio of two integers \(0.399999 \ldots\)

Find the area of the sector of a circle of radius 5 centimeters and central angle 2 radians (see Problem 42).

Does \((3,9)\) lie above or below the line \(y=3 x-1\) ?

\(f(x)=\frac{x^{3}+2}{x^{3}+1}\)

Let \(f(x+y)=f(x)+f(y)\) for all \(x\) and \(y\). Prove that there is a number \(m\) such that \(f(t)=m t\) for all rational numbers t. Hint: First decide what \(m\) has to be. Then proceed in steps, starting with \(f(0)=0, f(p)=m p\) for a natural number \(p\), \(f(1 / p)=m / p\), and so on.

Find the solution sets of the given inequalities. $$ \left|\frac{1}{x}-3\right|>6 $$

A regular polygon of \(n\) sides is inscribed in a circle of radius \(r\). Find formulas for the perimeter, \(P\), and area, \(A\), of the polygon in terms of \(n\) and \(r\).

Show that the equation of the line with \(x\)-intercept \(a \neq 0\) and \(y\)-intercept \(b \neq 0\) can be written as $$ \frac{x}{a}+\frac{y}{b}=1 $$

Find the solution sets of the given inequalities. $$ \left|2+\frac{5}{x}\right|>1 $$

Show that any rational number \(p / q\), for which the prime factorization of \(q\) consists entirely of \(2 \mathrm{~s}\) and \(5 \mathrm{~s}\), has a terminating decimal expansion.

Let \(f(x)=\left(x^{3}+3 x-5\right) /\left(x^{2}+4\right)\). (a) Evaluate \(f(1.38)\) and \(f(4.12)\). (b) Construct a table of values for this function corresponding to \(x=-4,-3, \ldots, 3,4\).

Find a positive rational number and a positive irrational number both smaller than \(0.00001\).

In Problems 45 -48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(4 x-5 y=8\) \(-3 x+y=5\) \(2 x+y=-10\)

Solve the given quadratic inequality using the Quadratic Formula. $$ x^{2}-4 x+4 \leq 0 $$

What is the smallest positive integer? The smallest positive rational number? The smallest positive irrational number?

Draw the graph of \(f(x)=x^{3}-5 x^{2}+x+8\) on the domain \([-2,5]\). (a) Determine the range of \(f\). (b) Where on this domain is \(f(x) \geq 0\) ?

Find a rational number between \(3.14159\) and \(\pi\). Note that \(\pi=3.141592 \ldots\)

Tides are often measured by arbitrary height markings at some location. Suppose that a high tide occurs at noon when the water level is at 12 feet. Six hours later, a low tide with a water level of 5 feet occurs, and by midnight another high tide with a water level of 12 feet occurs. Assuming that the water level is periodic, use this information to find a formula that gives the water level as a function of time. Then use this function to approximate the water level at \(5: 30\) P.M.

In Problems 45 -48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(5 x-2 y=5\) \(2 x+3 y=9\) \(2 x+3 y=6\)

Solve the given quadratic inequality using the Quadratic Formula. $$ 14 x^{2}+11 x-15 \leq 0 $$

Is there a number between \(0.9999 \ldots\) (repeating \(9 \mathrm{~s}\) ) and 1? How do you resolve this with the statement that between any two different real numbers there is another real number?

Circular motion can be modeled by using the parametric representations of the form \(x(t)=\sin t\) and \(y(t)=\cos t\). (A parametric representation means that a variable, \(t\) in this case, determines both \(x(t)\) and \(y(t)\).) This will give the full circle for \(0 \leq t \leq 2 \pi\). If we consider a 4 -foot- diameter wheel making one complete rotation clockwise once every 10 seconds, show that the motion of a point on the rim of the wheel can be represented by \(x(t)=2 \sin (\pi t / 5)\) and \(y(t)=2 \cos (\pi t / 5)\). (a) Find the positions of the point on the rim of the wheel when \(t=2\) seconds, 6 seconds, and 10 seconds. Where was this point when the wheel started to rotate at \(t=0\) ? (b) How will the formulas giving the motion of the point change if the wheel is rotating counterclockwise. (c) At what value of \(t\) is the point at \((2,0)\) for the first time?

The points \((2,3),(6,3),(6,-1)\), and \((2,-1)\) are corners of a square. Find the equations of the inscribed and circumscribed circles.

Classify each of the following as a PF (polynomial function), RF (rational function but not a polynomial function), or neither. (a) \(f(x)=3 x^{1 / 2}+1\) (b) \(f(x)=3\) (c) \(f(x)=3 x^{2}+2 x^{-1}\) (d) \(f(x)=\pi x^{3}-3 \pi\) (e) \(f(x)=\frac{1}{x+1}\) (f) \(f(x)=\frac{x+1}{\sqrt{x+3}}\)

Graph \(f(x)=(3 x-4) /\left(x^{2}+x-6\right)\) on the domain \([-6,6]\). (a) Determine the \(x\)-and \(y\)-intercepts. (b) Determine the range of \(f\) for the given domain. (c) Determine the vertical asymptotes of the graph. (d) Determine the horizontal asymptote for the graph when the domain is enlarged to the natural domain.

Show that the indicated implication is true. $$ |x-3|<0.5 \Rightarrow|5 x-15|<2.5 $$

Is \(0.1234567891011121314 \ldots\) rational or irrational? (You should see a pattern in the given sequence of digits.)

The circular frequency \(v\) of oscillation of a point is given by \(v=\frac{2 \pi}{\text { period }}\). What happens when you add two motions that have the same frequency or period? To investigate, we can graph the functions \(y(t)=2 \sin (\pi t / 5)\) and \(y(t)=\sin (\pi t / 5)+\) \(\cos (\pi t / 5)\) and look for similarities. Armed with this information, we can investigate by graphing the following functions over the interval \([-5,5]\) : (a) \(y(t)=3 \sin (\pi t / 5)-5 \cos (\pi t / 5)+2 \sin ((\pi t / 5)-3)\) (b) \(y(t)=3 \cos (\pi t / 5-2)+\cos (\pi t / 5)+\cos ((\pi t / 5)-3)\)

A belt fits tightly around the two circles, with equations \((x-1)^{2}+(y+2)^{2}=16\) and \((x+9)^{2}+(y-10)^{2}=16\) How long is this belt?

After being in business for \(t\) years, a manufacturer of cars is producing \(120+2 t+3 t^{2}\) units per year. The sales price in dollars per unit has risen according to the formula \(6000+700 t\). Write a formula for the manufacturer's yearly revenue \(R(t)\) after \(t\) years.

Show that the indicated implication is true. $$ |x+2|<0.3 \Rightarrow|4 x+8|<1.2 $$

Find two irrational numbers whose sum is rational.

We now explore the relationship between \(A \sin (\omega t)+\) \(B \cos (\omega t)\) and \(C \sin (\omega t+\phi)\). (a) By expanding \(\sin (\omega t+\phi)\) using the sum of the angles formula, show that the two expressions are equivalent if \(A=C \cos \phi\) and \(B=C \sin \phi .\) (b) Consequently, show that \(A^{2}+B^{2}=C^{2}\) and that \(\phi\) then satisfies the equation \(\tan \phi=\frac{B}{A}\). (c) Generalize your result to state a proposition about \(A_{1} \sin \left(\omega t+\phi_{1}\right)+A_{2} \sin \left(\omega t+\phi_{2}\right)+A_{3} \sin \left(\omega t+\phi_{3}\right)\). (d) Write an essay, in your own words, that expresses the importance of the identity between \(A \sin (\omega t)+B \cos (\omega t)\) and \(C \sin (\omega t+\phi) .\) Be sure to note that \(|C| \geq \max (|A|,|B|)\) and that the identity holds only when you are forming a linear combination (adding and/or subtracting multiples of single powers) of sine and cosine of the same frequency.

Show that the midpoint of the hypotenuse of any right triangle is equidistant from the three vertices.

Starting at noon, airplane A flies due north at 400 miles per hour. Starting 1 hour later, airplane \(B\) flies due east at 300 miles per hour. Neglecting the curvature of the Earth and assuming that they fly at the same altitude, find a formula for \(D(t)\), the distance between the two airplanes \(t\) hours after noon. Hint: There will be two formulas for \(D(t)\), one if \(0

Show that the indicated implication is true. $$ |x-2|<\frac{\varepsilon}{6} \Rightarrow|6 x-12|<\varepsilon $$

Find the best decimal approximation that your calculator allows. Begin by making a mental estimate. \((\sqrt{3}+1)^{3}\)