Chapter 1

Simplify the expression without using a calculator. Your answer should not have any radicals in it. $$\sqrt{\frac{1}{2}} \sqrt{\frac{1}{6}} \sqrt{\frac{1}{12}}$$

Find all real solutions of the equation exactly. $$6 x^{4}-7 x^{2}=3$$

Find an equation for the line satisfying the given conditions. Find a real number \(k\) such that (3,-2) is on the line \(k x-2 y+7=0\).

Find the center and radius of the circle whose equation is given. $$x^{2}+y^{2}+6 x-4 y-15=0$$

Simplify the expression without using a calculator. Your answer should not have any radicals in it. $$\sqrt{6}+\sqrt{2}(\sqrt{2}-\sqrt{3})$$

Solve the equation and check your answers. $$\frac{1}{2 t}-\frac{2}{5 t}=\frac{1}{10 t}-1$$

Find an equation for the line satisfying the given conditions. Find a real number \(k\) such that the line \(3 x-k y+2=0\) has \(y\) -intercept -3.

Find the center and radius of the circle whose equation is given. $$x^{2}+y^{2}+10 x-75=0$$

Simplify the expression without using a calculator. Your answer should not have any radicals in it. $$\sqrt{12}(\sqrt{3}-\sqrt{27})$$

Solve the equation and check your answers. $$\frac{1}{2}+\frac{2}{y}=\frac{1}{3}+\frac{3}{y}$$

If \(P\) is a point on a circle with center \(C\), then the tangent line to the circle at \(P\) is the straight line through \(P\) that is perpendicular to the radius \(C P\). In Exercises \(67-70\), find the equation of the tangent line to the circle at the given point. \(x^{2}+y^{2}=25\) at (3,4) [ Hint: Here \(C\) is (0,0) and \(P\) is \((3,4) ; \text { what is the slope of radius } C P ?]\)

Simplify the expression without using a calculator. Your answer should not have any radicals in it. \(\sqrt{u^{4}}(u \text { any real number })\) 68

Solve the equation and check your answers. $$\frac{2 x-7}{x+4}=\frac{5}{x+4}-2$$

If \(P\) is a point on a circle with center \(C\), then the tangent line to the circle at \(P\) is the straight line through \(P\) that is perpendicular to the radius \(C P\). In Exercises \(67-70\), find the equation of the tangent line to the circle at the given point. \(x^{2}+y^{2}=169\) at (-5,12)

Simplify the expression without using a calculator. Your answer should not have any radicals in it. \(\sqrt{3 x} \sqrt{75 x^{3}}(x \geq 0)\)

Solve the equation and check your answers. $$\frac{z+4}{z+5}=\frac{-1}{z+5}$$

If \(P\) is a point on a circle with center \(C\), then the tangent line to the circle at \(P\) is the straight line through \(P\) that is perpendicular to the radius \(C P\). In Exercises \(67-70\), find the equation of the tangent line to the circle at the given point. \((x-1)^{2}+(y-3)^{2}=5\) at (2,5)

Determine whether each point lies inside, or outside, or on the circle $$(x-1)^{2}+(y-3)^{2}=4$$ (a) (2.2,4.6) (b) (-.2,4.7) (c) (-.1,1.4) (d) (2.6,4.3) (e) (-.6,1.8)

Simplify, and write the given number without using absolute values. $$|3-14|$$

Solve the equation and check your answers. $$25 x+\frac{4}{x}=20$$

If \(P\) is a point on a circle with center \(C\), then the tangent line to the circle at \(P\) is the straight line through \(P\) that is perpendicular to the radius \(C P\). In Exercises \(67-70\), find the equation of the tangent line to the circle at the given point.) \(x^{2}+y^{2}+6 x-8 y+15=0\) at (-2,1)

Do the circles with the following equations intersect? $$(x-3)^{2}+(y+2)^{2}=25 \quad \text{and} \quad(x+3)^{2}+(y-2)^{2}=4$$ [Hint: Consider the radii and the distance between the centers. \(]\)

Simplify, and write the given number without using absolute values. $$|(-2) 3|$$

Solve the equation and check your answers. $$1-\frac{3}{x}=\frac{40}{x^{2}}$$

Find the equation of the circle. Center (3,3)\(;\) passes through the origin.

Simplify, and write the given number without using absolute values. $$3-|2-5|$$

Solve the equation and check your answers. $$\frac{2}{x^{2}}-\frac{5}{x}=4$$

If \(P\) is a point on a circle with center \(C\), then the tangent line to the circle at \(P\) is the straight line through \(P\) that is perpendicular to the radius \(C P\). In Exercises \(67-70\), find the equation of the tangent line to the circle at the given point.)Let \(A, B, C, D\) be nonzero real numbers. Show that the lines \(A x+B y+C=0\) and \(A x+B y+D=0\) are parallel.

Simplify, and write the given number without using absolute values. $$-2-|-2|$$

Find the equation of the circle. Center (1,2)\(;\) intersects \(x\) -axis at -1 and 3.

Simplify, and write the given number without using absolute values. $$\left|(-13)^{2}\right|$$

Solve the equation and check your answers. $$\frac{4 x^{2}+5}{3 x^{2}+5 x-2}=\frac{4}{3 x-1}-\frac{3}{x+2}$$

Worldwide motor vehicle production was about 60 million in 2000 and about 66 million in 2005 . (a) Let the \(x\) -axis denote time and the \(y\) -axis the number of vehicles (in millions). Let \(x=0\) correspond to 2000 . Fill in the blanks: the given data is represented by the points \((_-, 60)\) and \((5,-)\). (b) Find the linear equation determined by the two points in part (a). (c) Use the equation in part (b) to estimate the number of vehicles produced in 2004 . (d) If this model remains accurate, when will vehicle production reach 72 million?

Find the equation of the circle. Center (3,1)\(;\) diameter 2.

Simplify, and write the given number without using absolute values. $$-|-5|^{2}$$

Solve the equation and check your answers. $$\frac{x+3}{x-2}-\frac{3}{x+2}=\frac{20}{x^{2}-4}$$

Simplify, and write the given number without using absolute values. $$|\pi-\sqrt{2}|$$

Find the equation of the circle. Center (2,-6)\(;\) tangent to the \(y\) -axis.

Simplify, and write the given number without using absolute values. $$|\sqrt{2}-2|$$

The gross federal debt \(y\) (in trillions of dollars) in year \(x\) is approximated by $$ y=.79 x+3.93 \quad(x \geq 3) $$ where \(x\) is the number of years after \(2000 . \)Find the year in which the approximate federal debt is: \(\$ 14.2\) billion

The Missouri American Water Company charges residents of St. Louis County \(\$ 6.15\) per month plus \(\$ 2.0337\) per thousand gallons used." (a) Find the monthly bill when 3000 gallons of water are used. What is the bill when no water is used? (b) Write a linear equation that gives the monthly bill \(y\) when \(x\) thousand gallons are used. (c) If the monthly bill is \(\$ 22.42,\) how much water was used?

Find the equation of the circle. Endpoints of a diameter are (3,3) and (1,-1).

Simplify, and write the given number without using absolute values. $$|3-\pi|+3$$

The total health care expenditures \(E\) in the United States (in billions of dollars) can be approximated by $$ E=73.04 x+625.6 $$ where \(x\) is the number of years since \(1990 .^{\dagger}\) Determine the year in which health care expenditures are at the given level: \(\$ 1794.25\) billion

At sea level, water boils at \(212^{\circ} \mathrm{F}\). At a height of 1100 feet, water boils at \(210^{\circ} \mathrm{F}\). The relationship between boiling point and height is linear. (a) Find an equation that gives the boiling point \(y\) of water at a height of \(x\) feet. Find the boiling point of water in each of the following cities (whose altitudes are given). (b) Cincinnati, OH ( 550 feet) (c) Springfield, MO (1300 feet) (d) Billings, MT ( 3120 feet) (e) Flagstaff, AZ (6900 feet)

Find the equation of the circle. Endpoints of a diameter are (-3,5) and (7,-5).

Simplify, and write the given number without using absolute values. $$|4-\sqrt{2}|-5$$

The total health care expenditures \(E\) in the United States (in billions of dollars) can be approximated by $$ E=73.04 x+625.6 $$ where \(x\) is the number of years since \(1990 .^{\dagger}\) Determine the year in which health care expenditures are at the given level: \(\$ 1940.3\) billion

According to the Center of Science in the Public Interest, the maximum healthy weight for a person who is 5 feet 5 inches tall is 150 pounds, and the maximum healthy weight for someone 6 feet 3 inches tall is 200 pounds. The relationship between weight and height here is linear. (a) Find a linear equation that gives the maximum healthy weight \(y\) for a person whose height is \(x\) inches over4 feet 10 inches. (Thus \(x=0\) corresponds to 4 feet \(10 \text { inches, } x=2 \text { to } 5 \text { feet, etc. }) \quad U_{S \in}\). (b) What is the maximum healthy weight for a person whose cise height is 5 feet? 6 feet? (c) How tall is a person who is at a maximum healthy.

One diagonal of a square has endpoints (-3,1) and \((2,-4) .\) Find the endpoints of the other diagonal.