Chapter 1

Solve each problem analytically, and support your solution graphically. Acid Mixture A chemist needs \(10 \%\) hydrochloric acid for an experiment. How much \(5 \%\) acid should be mixed with 60 milliliters of \(20 \$ 6\) acid to obtain a \(10 \%\) solution?

Graph each set of numbers on a number line. $$\\{-6,-5,-4,-3,-2\\}$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$2 x-5=x+7$$

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((0,5)\) and \((10,0)\)

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=\frac{1}{2} x$$

Solve each problem analytically, and support your solution graphically. Alcohol Mixture How many gallons of pure alcohol should be mixed with 20 gallons of a \(15 \%\) alcohol solution to obtain a mixture that is \(25 \%\) alcohol?

Graph each set of numbers on a number line. $$\left\\{-0.5,0.75, \frac{5}{3}, 3.5\right\\}$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$9 x-17=2 x+4$$

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((0,-8)\) and \((4,0)\)

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=-\frac{2}{3} x$$

Solve each problem analytically, and support your solution graphically. Alcohol Mixture \(\quad\) A chemist wishes to strengthen a mixture from \(10 \%\) alcohol to \(30 \%\) alcohol. How much pure alcohol should be added to 7 liters of the \(10 \%\) mixture?

Graph each set of numbers on a number line. $$\left\\{-0.6, \frac{9}{8}, 2.5, \frac{13}{4}\right\\}$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$0.01 x+3.1=2.03 x-2.96$$

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. $$\begin{array}{c|c} x & y \\ -7 & -44 \\ -6 & -36 \\ -5 & -28 \\ -4 & -20 \end{array}$$

Graph each line. Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). $$f(x)=-3$$

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=x^{2}$$

Solve each problem analytically, and support your solution graphically. Saline Solution Mixture How much water should be added to 8 milliliters of 6\% saline solution to reduce the concentration to \(4 \%\) saline?

Explain the distinction between a rational number and an irrational number.

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$0.04 x+2.1=0.02 x+1.92$$

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. $$\begin{array}{|c|c|} \hline x & y \\ \hline-2.4 & 5.2 \\ 1.3 & -24.4 \\ 1.75 & -28 \\ 2.98 & -37.84 \end{array}$$

Graph each line. Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). $$f(x)=5$$

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=-x^{2}$$

Solve each problem analytically, and support your solution graphically. Acid Mixture How much water should be added to 20 liters of an \(18 \%\) acid solution to reduce the concentration to \(15 \%\) acid?

Using her calculator, a student found the decimal 1.414213562 when she evaluated \(\sqrt{2} .\) Is this decimal the exact value of \(\sqrt{2}\) or just an approximation of \(\sqrt{2} ?\) Should she write \(\sqrt{2}=1.414213562\) or \(\sqrt{2} \approx 1.414213562 ?\)

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$-(x+5)-(2+5 x)+8 x=3 x-5$$

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. $$\begin{array}{c|c} x & y \\ \hline 2 & -5 \\ 3 & -8 \\ 4 & -11 \\ 5 & -14 \end{array}$$

Graph each line. Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). $$f(x)=2.5$$

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(5,1),(3,2),(4,9),(7,6)\\}$$

Solve each problem analytically, and support your solution graphically. Antifreeze Mixture An automobile radiator holds 16 liters of fluid. There is currently a mixture in the radiator that is \(80 \%\) antifreeze and \(20 \%\) water. How much of this mixture should be drained and replaced by pure antifreeze so that the resulting mixture is \(90 \%\) antifreeze?

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(2,3)$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$-(8+3 x)+5=2 x+3$$

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. $$\begin{array}{c|c} x & y \\ -1.1 & 1.5 \\ -1.0 & 2.0 \\ -0.9 & 2.5 \\ -0.8 & 3.0 \end{array}$$

Graph each line. Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). $$f(x)=1.25$$

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(8,0),(5,4),(9,3),(3,8)\\}$$

Solve each problem analytically, and support your solution graphically. Antifreeze Mixture An automobile radiator contains a 10 -quart mixture of water and antifreeze that is \(40 \%\) antifreeze. How much should the owner drain from the radiator and replace with pure antifreeze so that the liquid in the radiator will be \(80 \%\) antifreeze?

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(1,2)$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$\frac{2 x+1}{3}+\frac{x-1}{4}=\frac{13}{2}$$

Graph each line by hand. Give the \(x\) - and y-intercepts. $$x-y=4$$

Graph each line. Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). $$x=2$$

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(1,6),(2,6),(3,6)\\}$$

involve octane rating of gasoline, \(a\) measure of its antiknock qualities. In one measure of octane, a standard fuel is made with only two ingredients: heptane and isooctane. For this type of fuel, the octane rating is the percent of isooctane. An actual gasoline blend is then compared with a standard fuel. For example, a gasoline with an octane rating of 98 has the same antiknock properties as a standard fuel that is \(98 \%\) isooctane. Octane Rating of Gasoline How many gallons of 94-octane gasoline should be mixed with 400 gallons of 99-octane gasoline to obtain a mixture that is 97 -octane?

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-3,2)$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$\frac{x-2}{4}+\frac{x+1}{2}=1$$

Graph each line by hand. Give the \(x\) - and y-intercepts. $$x+y=4$$

Graph each line. Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). $$x=-3$$

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(-10,5),(-20,5),(-30,5)\\}$$

involve octane rating of gasoline, \(a\) measure of its antiknock qualities. In one measure of octane, a standard fuel is made with only two ingredients: heptane and isooctane. For this type of fuel, the octane rating is the percent of isooctane. An actual gasoline blend is then compared with a standard fuel. For example, a gasoline with an octane rating of 98 has the same antiknock properties as a standard fuel that is \(98 \%\) isooctane. Octane Rating of Gasoline How many gallons of 94-octane gasoline should be mixed with 400 gallons of 99-octane gasoline to obtain a mixture that is 97 -octane?

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-4,3)$$

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$\frac{1}{2}(x-3)=\frac{5}{12}+\frac{2}{3}(2 x-5)$$

Graph each line by hand. Give the \(x\) - and y-intercepts. $$3 x-y=6$$