About this edition

This web edition of Preparing for Algebra and Statistics by Matt Teachout is our adaptation of the author's book. Below are the 261 changes we made to individual passages and figures, each listed once, under the section where it appears.

We also made changes across the whole book, such as converting it to accessible web pages, rebuilding the math, describing the figures and adding answer checks to some examples. We added 46 narrated lesson videos and worked solutions to 203 examples. The narration is a synthetic (computer-generated) voice. Those changes are not listed one by one. All of these changes are ours. They do not mean that the author has reviewed or approved them. The license and the attribution to the author are on the book's home page. The original book is free at matt-teachout.org.

A change listed here does not mean that the rest of the book is free of errors. If you find an error, use Report a content issue at the bottom of this page.

Changes in the front matter

Front Matter: Introduction

The introduction said the answer keys to the odd problems are in appendix A. This web edition has no appendix A: the answers appear under the problems, behind a Check answer button. The sentence now says so, and says that Section 7D and problem 9 of the Chapter 2 Review have none.

Changes in Chapter 1: Formulas and Whole Numbers

1A Formulas with Adding and Subtracting Whole Numbers

Edition note: This note is new. The opening example describes a pasture 461 feet by 345 feet and Example 3 a field 451 feet by 345 feet, with perimeters of 1,612 and 1,592 feet; the note says to use the dimensions given in each example, because the book does not say whether the different lengths were intended.
The carrying rule said "Whenever we go over 10 in a place value"; it now says "Whenever the sum in a place value is 10 or more", because a sum of exactly 10, such as 1 plus 9, also needs a carry.
Problem 37: The key printed 608 with a dollar sign; it now prints 608, because 2101 minus 1493 is a plain subtraction with no money in it. The number was already correct.
Problem 39: The key printed 2,848 with a dollar sign; it now prints 2,848, because 3,721 minus 873 is a plain subtraction with no money in it. The number was already correct.
The opening paragraph printed "add, subtract multiply or divide"; it now reads "add, subtract, multiply or divide", with the missing comma added.
In the worked subtraction of 268 from 513, one step printed "2 hundred" where the rest of the step says hundreds; it now reads 4 hundreds minus 2 hundreds equals 2 hundreds.
Problem 47 printed "Empire State building" and "Chrysler building"; it now prints Empire State Building and Chrysler Building, because Building is part of each name.
Problem 49 printed "Taxi driver" with a capital T; it now prints "taxi driver", because it is not a name or the start of a sentence.
Problem 50 printed "three day period"; it now prints "three-day period", with a hyphen, because three-day describes the period.
Problem 48 stated the heights as current facts; it now says they are approximate, that 1063 feet is the height the Eiffel Tower formerly measured including its antenna, and that the Statue of Liberty is 305 feet from the ground to the tip of its torch, so it asks how much taller the tower was. The tower's measured height has since changed, and the numbers to subtract are the same as before.

1B Formulas with Multiplying Whole Numbers and Positive Exponents

The shortcut for multiplying numbers with zeros said "Multiply the non-zero numbers and then add on all the zeros"; it now says to set aside only the zeros at the end of each factor, multiply what is left, then attach all of those zeros. Zeros inside a number cannot be set aside, because they hold place value: 2,005 times 8 is 16,040.
The definition said an exponent "tells you how many times you should multiply the base times itself"; it now says a positive whole-number exponent is the number of copies of the base used as factors, so an exponent of 1 means the base alone. This matches the book's example, where 7 to the fourth power is four 7s multiplied together, which takes three multiplications, not four.
The practice directions said to perform the indicated operation for problems 1 to 48; they now say 1 to 54, because problems 49 to 54 are also operation problems and the word problems start at problem 55.
The sentence about the volume of a box printed "length width and height"; it now reads "length, width and height", with the missing comma added.
The reminder about multiplication printed "repeat addition"; it now says "repeated addition".
Problem 57 printed "square shaped garden"; it now prints "square-shaped garden", with a hyphen, because square-shaped describes the garden.

1C Formulas with Division

The rule said "Any number divided by itself is equal to 1"; it now says any non-zero number, because 0 divided by 0 has no single answer: every number times 0 is 0.
Edition note: This note is new. The section divides with remainders and uses the z-score formula without stating their conditions; the note says the divisor cannot be zero, a remainder is smaller than the divisor, the standard deviation must be positive and in the same units as the data, and treating a z-score greater than 2 as unusual is the rule for these exercises, not a general statistical test.
The original book prints the heading of this section's practice problems as Practice Problems Section IC, with a capital letter I where the digit 1 belongs. This edition prints Practice Problems Section 1C, which matches the section number.

1D Formulas with Square Roots and Order of Operations

Publisher clarification: This clarification is new. The book defines a square root as a number we can square to get the number under the root sign, which fits both 7 and −7 for 49; the clarification says the square-root sign means the nonnegative root, the square root of 0 is 0, and a negative number has no real square root.
Publisher clarification: This clarification is new. The z-score formula for a sample mean was given without its conditions; the clarification says the sample size must be a positive whole number, the standard deviation must be positive, and the means and standard deviation must use the same units, so the z-score has no unit.
Publisher clarification: This clarification is new. The distance formula was given without saying anything about units; the clarification says both coordinates must use the same unit, the distance is never negative and is in that unit, and because these exercises name no unit, their answers are in coordinate units.
The practice directions said to perform the indicated operation for problems 1 to 29; they now say 1 to 35, because problems 30 to 35 are also expressions to evaluate. The z-score problems are 36 to 39 and the distance problems are 40 to 43.
The worked order of operations example printed "lets look at"; it now prints "let's look at", with the apostrophe.
The first step in the Order of Operations list printed "Parenthesis"; it now prints "Parentheses", the plural, because the step covers every kind of grouping symbol shown with it.

Chapter 1 Review

The practice directions said to perform the indicated operation for problems 1 to 46; they now say 1 to 53, because problems 47 to 53 are also operation problems and the word problems start at problem 54.

Changes in Chapter 2: Formulas and Decimals

2A Rounding, Comparing, Adding and Subtracting Decimals

Problem 22 printed "Round 0.053289 to the thousandths", which left the instruction unfinished; it now ends "to the thousandths place." The number to round is unchanged.
Problem 24 printed "Round 8.00374 to the hundredths", which left the instruction unfinished; it now ends "to the hundredths place." The number to round is unchanged.
Problem 31 printed "to the nearest dollar. ones place)." with a stray period and no opening parenthesis; it now reads "to the nearest dollar (ones place)." The amount and the answer, 482 dollars, are unchanged.
The instruction to add or subtract in order from left to right said it covered problems 56 to 61; it now says 63 to 68, the numbers those problems actually have (the subtraction problems before them are 48 to 62). No problem or answer changed.
Publisher clarification: Added after the comparison showing that 0.05 is larger than 0.0035, where the original still says this means the P-value is significantly low. It explains that the comparison is correct, but a P-value is only called statistically significant against a significance level chosen in advance, such as 0.05, using a suitable test whose conditions hold.
In the example of subtracting 3.2751 from 9, the sentence that introduces it had no period and ran into "First let's rewrite the 9"; a period was added.

2B Formulas with Exponents and Multiplying Decimals

The instruction "simplify any exponents and then multiply" said it covered problems 1 to 33; it now says 1 to 57, because problems 34 to 57 are the same kind of multiplication and the next instruction, on circle areas, starts at problem 58. No problem or answer changed.
Problem 65 printed "the volume of a cube with if one side", with an extra word; it now reads "the volume of a cube if one side of the cube is 1.2 cm long." The answer is unchanged.
Problem 61: The answer printed 26.4074 with no unit; it now reads 26.4074 sq cm. The radius is 2.9 cm, so the area is in square centimeters; the number itself is unchanged.
The sentence "Recall that exponents are repeated multiplication" now reads "For the positive whole-number exponents used here, exponents mean repeated multiplication." Repeated multiplication describes positive whole-number exponents like the 2 and 3 in the examples, not zero, negative or fractional exponents.
The instruction before the cube volume problems said "Find the volume of cube"; it now says "Find the volume of a cube".

2C Formulas with Dividing Decimals

Publisher clarification: Added after the paragraph that says the divisor must be a whole number, which is unchanged. It explains that multiplying the dividend and a nonzero divisor by the same power of ten keeps the quotient the same, so a whole-number divisor is a step of this long-division method rather than a rule of division, and that division by zero is undefined.
Publisher clarification: Added after the bigger and smaller memory aid for multiplying and dividing by powers of ten. It says the aid applies to positive numbers, that zero stays zero, and that for negative numbers the decimal point moves the same way but multiplying makes the number more negative, not larger.
Publisher clarification: The sentence before the z-score practice problems said they solve the problem from the beginning of the section, but that example uses a standard deviation of 2.5 pounds while problem 62 uses 1.6 pounds; the sentence now says to apply the z-score formula to the practice values. This note was added to point out the difference and to say that the standard deviation must be positive and all three values must be in the same units.
Problem 65: The answer printed 0.523, with the 3 repeating, and no unit; it now adds cubic yards, because the radius and height are given in yards and the answer is a volume. The number is unchanged.
Problem 48, which asks you to convert three fifths into a decimal, ended without a period; a period was added. Nothing else changed.
Problem 52, which asks you to convert seven twenty-fourths into a decimal, ended without a period; a period was added. Nothing else changed.
Problem 56, which asks you to convert nineteen thirtieths into a decimal, ended without a period; a period was added. Nothing else changed.
Problem 59 asks how many months it will take to save for a down payment but ended that question with a period; it now ends with a question mark. The amounts and the answer, 17 months, are unchanged.
The note about zero and one said zero divided by any number is zero; it now says any nonzero number, because zero divided by zero is undefined, as the same note says of any division by zero.

2D Scientific Notation

Publisher clarification: The paragraph that introduces the P-value said a P-value is often very close to zero and called 0.000000000173 a difficult number to interpret; it now says P-values lie from zero to one, that this example is very close to zero, and that it is a long decimal. This note was added to explain that scientific notation changes how a P-value is written, not its value, and that the notation alone says nothing about statistical significance.
Publisher clarification: The definition of scientific notation now begins "For the positive numbers used here", and this note was added after it. The note says the first number is at least 1 and less than 10, the exponent is any integer including zero, a negative number uses a first number whose absolute value is in that range, and zero is written separately.
Problem 49: The answer still says small numbers have a negative exponent and large numbers a positive one, and now adds that, with a first number from 1 up to but not including 10, a negative exponent gives a value between zero and one, an exponent of zero gives a value from 1 up to but not including 10, and a positive exponent gives 10 or more. The original answer did not cover an exponent of zero.
Problem 55: The answer printed 8 times 10 to the power of negative 5 with no unit; it now adds meters, the unit the problem gives for the diameter of the hair (0.00008 meters). The number is unchanged.
Problem 57: The answer said the first number in scientific notation must be between 1 and 10 and that 0.08 is not; it now says the first number must be at least 1 and less than 10, so 1 is allowed and 10 is not, and 0.08 fails because it is less than 1. The reason is the same; the new wording makes clear which end points are allowed.
The instruction for problems 1 to 36, "Convert the following numbers into scientific notation", ended without a period; a period was added.

2E Significant Figure Rules

Publisher clarification: Added after the list of significant figure rules, which is unchanged. It says to follow the book's rule that zeros at the end of a whole number, as in 31,000, are not significant in these exercises, but that in real measurements such zeros may be significant and the number alone does not show it.
Publisher clarification: Added under the heading on rounding rules with significant figures. It says to keep all digits until the final answer and round only once, that a whole number counts as having zero decimal places in the rule for adding and subtracting, and that decimal zeros the rule calls for must be kept in a rounded answer.
In the rule for multiplying or dividing, the explanation of the example 12.831 times 17.4, which rounds to 223, opened a parenthesis and never closed it; the closing parenthesis was added.
Problem 49: The answer printed 1.9; it is now 31, because 7.631 divided by 0.25 equals 30.524, and rounding to two significant figures (0.25 has only two) gives 31.

2F Estimating Square Roots and Order of Operations with Decimals

Publisher clarification: Added after the list of perfect squares. It explains that the square root symbol means the nonnegative root (the square root of 49 is 7, not negative 7), that zero is also a perfect square, and that any close estimate ending in the tenths place is accepted in this section, as in the answer key.
The instruction "Perform the indicated operations" gave the problem numbers as 16 to 21; it now gives 34 to 44, the numbers of those problems on the page (problem 45 starts the distance formula problems). No problem or answer changed.
In the paragraph that revisits the distance formula, the second point ran straight into the next word and the text said "on a plane"; it now has a space there and says "in the Cartesian plane", the coordinate plane the formula works in.
In Example 1, the square root of 59 ran straight into the next word ("as a decimal"); the instruction now has a space and reads "using a decimal that ends in the tenths place". The solution is unchanged.
In the worked order of operations example, the step 4.7 minus 4.1 equals 0.6 ran into the next sentence ("Now do the exponents"); a period was added after 0.6.

2G Statistics Applications with Decimals

The definition of the mean printed "The first statistic we will learn to calculate the mean average", missing the word "is", and the summation symbol ran into the words "means to add". The missing word and the space were added, and the sentence now says the symbol means to add the data values.
In Example 1, about a runner training for a marathon, the data description said "has ran"; it now says "has run", the correct verb form.
The instruction to round money to the nearest cent now applies to amounts given in dollars, and a sentence was added saying exact medians and ranges may leave off trailing zeros. Problem 19 gives coffee prices in cents, so the instructions now say to report its mean to three decimal places in cents, matching the answer key (95.179 cents).
Problem 19 spelled the coffee "Columbian"; it now reads "Colombian", the spelling for coffee from Colombia. The prices and the answer are unchanged.
Problem 20 printed "degrees fahrenheit" in lowercase; Fahrenheit is now capitalized because it is the name of a temperature scale.

Chapter 2 Review

The rounding summary printed the sentence "If it is 5 or above, round up" twice in a row; the repeat was removed.
The significant figures summary now says digits instead of numbers, corrects "Zero's" to "Zeros", and labels the rule that zeros at the end of a whole number are not significant as this chapter's classroom convention. It also adds that trailing zeros after a decimal point are significant, as the chapter's earlier rules say, and that for measured data the stated precision decides.
The dividing decimals summary now says the divisor must be nonzero, and "Remember the devisor must be whole number" (misspelled and missing a word) now reads "For this long-division method, make the divisor a whole number", because a whole-number divisor is a step of the method, not a rule of division.
The scientific notation summary said "a number between 1 and 10 times a power of 10" and that small numbers have a negative power and large numbers a positive one; it now says the first number is at least 1 and less than 10 and the power is an integer. It also says exactly which positive numbers get a negative, zero or positive exponent, since the old wording left out an exponent of zero.
The instruction for problems 57 to 61 said "tenth's place" with an apostrophe; it now says "tenths place".

Changes in Chapter 3: Formulas and Fractions

3A Fractions and Mixed Number Conversions

The pizza example did not say that the pizzas were the same size or the pieces equal, and it called 20 total pieces a common mistake. It now says both, notes that there are 20 pieces across the two pizzas, and explains that eating 13 pieces is 13/10 because the denominator counts the 10 equal pieces in one pizza.

3B Simplifying and Equivalent Fractions

The text said 1/2 is in simplest form because it is simpler than 20/40, and that multiplying the numerator and denominator by the same number gives an equal fraction. It now defines simplest form as having no common positive whole-number factor greater than 1, and says the number you multiply by must not be zero, because multiplying by zero gives 0/0, which is not equal to the fraction.
The text said the key is dividing (canceling) common factors and that we divided out any numbers that go into both the numerator and denominator. It now says to divide the complete numerator and denominator by the same nonzero common factor, because canceling only part of either one, or dividing by zero, does not give an equal fraction; it also defines simplest form as no common positive whole-number factor greater than 1.
In the 24/72 example, the text asked for numbers that go into both 24 and 72 and said to divide the top and bottom by the same number. It now asks for positive whole-number factors greater than 1 and says to divide the complete numerator and denominator by the same nonzero common factor, because 1 divides every whole number and dividing by zero is undefined.
After 3/9 is reduced to 1/3, the text said there are no numbers that go into 1 and 3 besides 1. It now says no positive whole-number factor greater than 1 divides both 1 and 3, matching the definition of simplest form given earlier in this section.

3C Decimal and Fraction Conversions

The text said some fractions convert into "a repeating decimals"; it now says "repeating decimals", a grammar fix.
The text said the whole part of a mixed number is the digits to the left of the decimal point, then said to add the decimal for the fraction part to it; this is now stated for positive mixed numbers. For a negative mixed number the method fails: negative 2 and 1/4 is negative 2.25, but adding 0.25 to negative 2 gives negative 1.75.
Three wording fixes in the paragraph on converting decimals to fractions: "stem on" now reads "stem from"; the reason 7/10 is in simplest form now reads that no positive whole-number factor greater than 1 divides both 7 and 10, instead of that no numbers go into both except 1; and "The fraction ends in the thousandths place" now reads "The decimal ends", because 0.024 is a decimal, not a fraction.

3D Formulas with Multiplying and Dividing Fractions

The text said 2/15 is in simplest form since there are no common factors between 2 and 15. It now says no positive whole-number factor greater than 1 divides both, because 1 is a common factor of every pair of whole numbers.
The text said exponents just mean repeated multiplication; it now says positive integer exponents mean repeated multiplication, because an exponent of zero, a negative exponent or a fractional exponent is not repeated multiplication.
The text said that to divide fractions we multiply by the reciprocal and flip the divisor. It now says this is for dividing by a nonzero fraction and flipping a nonzero divisor, because zero has no reciprocal and division by zero is undefined.
In the 9/10 divided by 3/8 example, the text said dividing the 6 gives the answer in simplest form. It now says dividing both the numerator and the denominator by 6, which is the step the worked equation shows.
The directions before problems 38 to 41 said to use the cone volume formula for all four problems, but problem 41 asks for the circumference of a circular track. The directions now say to use the volume formula for the cone problems and, for the circumference problem, the formula given in that problem, with the same approximation for pi.
Problem 35 stated that an estimated 9/10 of children in the U.S. have had their vaccinations, then asked about a small town. It now asks you to suppose that 9/10 of the children in the town have had them, and asks how many are estimated not to have had them, because a national rate need not hold in one town and the answer depends only on the fraction the problem gives.
Problem 13: The key printed only the mixed number 1 and 1/2. It now prints the simplest fraction 3/2 and the mixed number 1 and 1/2, because the directions ask for the answer in simplest form and, when it is an improper fraction, as a mixed number also.
Problem 19: The key printed only the mixed number 4 and 1/2. It now prints the simplest fraction 9/2 and the mixed number 4 and 1/2, because the directions ask for the answer in simplest form and, when it is an improper fraction, as a mixed number also.
Problem 29: The key printed only the mixed number 1 and 1/3. It now prints the simplest fraction 4/3 and the mixed number 1 and 1/3, because the directions ask for the answer in simplest form and, when it is an improper fraction, as a mixed number also.
Problem 31: The key printed only the mixed number 1 and 2/3. It now prints the simplest fraction 5/3 and the mixed number 1 and 2/3, because the directions ask for the answer in simplest form and, when it is an improper fraction, as a mixed number also.
Problem 33: The key printed only the mixed number 26 and 2/3. It now prints the simplest fraction 80/3 and the mixed number 26 and 2/3, because the directions ask for the answer in simplest form and, when it is an improper fraction, as a mixed number also.
Problem 35: The key gave only 21,060 children, the number estimated to have had their vaccinations. It now also gives the second number the problem asks for, 2,340 children estimated not to have had them, since 23,400 minus 21,060 is 2,340.

3E Formulas with Multiplying and Dividing Mixed Numbers

In the example of 5 and 5/8 divided by 6 and 2/3, the rule for changing a mixed number to an improper fraction (multiply the whole part by the denominator, then add the numerator) is now stated for positive mixed numbers. Applied to negative 5 and 5/8, the rule would give negative 35/8 instead of negative 45/8.
In the directions for the cupcake recipe problems, "need" now reads "needs", a grammar fix.
The directions for the practice problems asked for answers as a simplified proper fraction or a mixed number. They now also allow a whole number, because some answers in this set are whole numbers, such as 14 in problem 1 and 4 in problem 19.

3F Unit Conversions

In 4 places (Example 3 and problems 11, 12 and 29), the factor printed as 1 mph equals 1.47 fps now reads 1 mph is approximately 1.47 fps. The exact factor is 5280 feet per mile divided by 3600 seconds per hour, which is 22/15, or about 1.4667, so 1.47 is rounded.
Edition note: The original had no note here. The added note explains that rounded factors such as 1.47 fps per mph and 4.2 cups per liter give approximate results; for example, 60 mph is exactly 88 fps, but 60 times 1.47 gives 88.2 fps.
The opening said that if you know about fractions, you can convert to any unit you need to. It now says fractions give a systematic way to make the multiplicative unit conversions this section teaches, so the claim matches what the section covers.
In the cups to liters example, the text said to use the equal quantities to make a unit fraction. It now says to use the supplied rounded conversion factor, because 1 liter is only approximately 4.2 cups, as the example itself states.
After the cups to liters example, the text said we canceled the cups and were left with units. It now says we were left with liters, the unit of the answer, 2.5 liters.
In problem 9, the factor printed as 1 inch is approximately 2.54 centimeters now reads 1 inch equals 2.54 centimeters, because the inch is defined as exactly 2.54 centimeters.
In problem 10, the factor printed as 1 inch is approximately 2.54 centimeters now reads 1 inch equals 2.54 centimeters, because the inch is defined as exactly 2.54 centimeters.
In problem 15, the factor printed as 1 kg equals 2.2 pounds now reads 1 kg is approximately 2.2 pounds, because 1 kilogram is about 2.2046 pounds.
In problem 16, the factor printed as 1 kg equals 2.2 pounds now reads 1 kg is approximately 2.2 pounds, because 1 kilogram is about 2.2046 pounds.
In problem 17, the factor printed as 1 kph equals 0.62 mph now reads 1 kph is approximately 0.62 mph, because 1 kilometer per hour is about 0.6214 miles per hour.
In problem 18, the factor printed as 1 kph equals 0.62 mph now reads 1 kph is approximately 0.62 mph, because 1 kilometer per hour is about 0.6214 miles per hour.
In problem 31, the factor printed as 1 kg equals 2.2 pounds now reads 1 kg is approximately 2.2 pounds, because 1 kilogram is about 2.2046 pounds.
In problem 32, the factor printed as 1 kph equals 0.62 mph now reads 1 kph is approximately 0.62 mph, because 1 kilometer per hour is about 0.6214 miles per hour.
In problem 12, a period is added after "Write your answer as a decimal", before the conversion factor in parentheses.
In problem 17, the extra spaces just inside the parentheses around the conversion factor are removed, a spacing fix.
In problem 18, the extra spaces just inside the parentheses around the conversion factor are removed, a spacing fix.
In problem 21, "milliLiters" and "Liters" now read "milliliters" and "liters", the standard lowercase spelling of these unit names.
In problem 22, "milliLiters" and "Liters" now read "milliliters" and "liters", the standard lowercase spelling of these unit names.
In problem 26, the directions said to write the answer as a mixed number; they now say a whole number, because converting 2.75 tons at 2000 pounds per ton gives a whole number of pounds.

3G Adding and Subtracting Fractions

In the pizza example, the text said the denominator is how many pieces 1 piece is cut up into. It now says the denominator is how many equal pieces one whole is cut up into, which matches the definition given at the start of this section.
Problem 13: The key printed 44/77; it is now 47/77, because 2/11 plus 3/7 is 14/77 plus 33/77, which is 47/77.
Problem 31: The key printed 1/50; it is now 31/50, because 8/25 plus 3/10 is 16/50 plus 15/50, which is 31/50.
In the pizza example, the comma in "We did not cut the pizza into 20 pieces, it was cut up into 10 pieces" is now a semicolon, because the two parts are complete sentences.

3H Adding and Subtracting Mixed Numbers

In the opening custard problem, "deserts" is corrected to "desserts".
The sentence defining a mixed number now reads "a mixed number is a whole number added to a fraction"; the word "a" was missing before "whole number".
In the example of 6 and 4/5 plus 2 and 2/3, "add the numerators and keeping the denominator" now reads "add the numerators and keep the denominator", a grammar fix.
A comma is added after "For example" in the sentence that introduces the example of 6 and 4/5 plus 2 and 2/3.
In problem 35, "Liters" now reads "liters" in both places, the standard lowercase spelling of the unit name.

Chapter 3 Review

In problem 45, the factor printed as 1 inch is approximately 2.54 centimeters now reads 1 inch equals 2.54 centimeters, because the inch is defined as exactly 2.54 centimeters.
In problem 46, the factor printed as 1 kg equals 2.2 pounds now reads 1 kg is approximately 2.2 pounds, because 1 kilogram is about 2.2046 pounds.
In problem 47, the factor printed as 1 kph equals 0.62 mph now reads 1 kph is approximately 0.62 mph, because 1 kilometer per hour is about 0.6214 miles per hour.

Changes in Chapter 4: Formulas and Negative Numbers

4A Negative Quantities and Absolute Value

The paragraph on absolute value said an absolute value is always "a positive quantity"; it now says "a nonnegative quantity", because the absolute value of 0 is 0, which is not positive.
Problem 5 printed "The amount of people"; it now prints "The number of people", because people are counted, not measured.
Problem 10 printed "The amount of people"; it now prints "The number of people", because people are counted, not measured.
Problem 13 printed "The number of cases of the flu have increased"; it now prints "has increased", because the subject, number, is singular.

4B Adding Negative Quantities

The opening paragraph about a stock price printed "over that last few weeks"; it now prints "over the last few weeks".
The rule for adding two numbers with the same sign said to "add the numbers and keep the sign"; it now says to add their absolute values and keep the common sign. This matches the worked example, where −10 plus −7 is found by adding 10 and 7 and keeping the negative sign.
The rule for adding numbers with opposite signs said to "subtract the numbers and keep the sign of the larger"; it now says to subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value, and that equal absolute values give 0. In −13 plus 8, the larger number is 8, yet the sum, −5, takes the sign of −13, and practice sums such as −17 plus 17 are 0.
The rules for adding fractions and decimals said to "add the numbers and keep the sign" when the signs are the same and to "subtract the numbers and keep the sign of the larger" when they are opposite; they now use absolute values in the same way as the corrected rules earlier in this section, including that equal absolute values give 0.
Problem 37 was missing a word in its first sentence; it now reads "is balancing her checkbook".

4C Subtracting Negative Quantities

The definition said opposites "have the same number, but have the opposite sign"; it now says opposites have the same absolute value, nonzero opposites have opposite signs, and zero is its own opposite, because 0 has no sign.
In the worked example −9 minus −10, the reminder of the addition rule said to "subtract the numbers and keep the sign of the larger"; it now gives the corrected rule: subtract the smaller absolute value from the larger, keep the sign of the number with the larger absolute value, and give 0 when the absolute values are equal. The example's answer, positive 1, is unchanged.
In the example −13 plus 13, the text said to "subtract the numbers and keep the sign of the larger", but neither number has the larger absolute value; it now says to compare the absolute values and that equal absolute values cancel to zero. The sum is still 0.
Problem 38 said to subtract 45 from −8 to find how many points the team gave up, but that subtraction gives −53, while a number of points is never negative; it now says the subtraction gives the negative contribution of the points given up, and its absolute value is the number of points.
Problem 38: The key printed −53 points; it now says the negative contribution is −53 points, so the team gave up 53 points. A count of points cannot be negative, and 45 minus 53 gives the stated plus/minus value of −8.

4D Multiplying and Dividing Negative Quantities

This note said that when two numbers with opposite signs are added, the answer "can be negative or positive depending on which is larger"; it now says negative, positive, or zero, depending on the absolute values, because opposites such as −8 and 8 add to 0 and the sign follows the larger absolute value, not the larger number.
This note said 0 divided by any number is 0; it now adds "when the divisor is nonzero", because 0 divided by 0 is undefined, as the same note says of any division by zero.

Chapter 4 Review

The summary said that to add two negative numbers you "add the numbers and keep the sign"; it now says to add their absolute values and keep the negative sign. For example, −7 plus −12 is found by adding 7 and 12 and keeping the negative sign, which gives −19.
The summary said that to add a negative and a positive number you "subtract the numbers and keep the sign of the larger"; it now says to subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value, with a sum of 0 when the absolute values are equal. In −13 plus 8, the larger number is 8, yet the sum, −5, takes the sign of −13.
Problem 5 printed "The amount of people"; it now prints "The number of people", because people are counted, not measured.
Problem 13: The key printed −2.5084; it now prints −2.5804, because the sum of −0.0014 and −2.579 is −2.5804. Two digits had been switched.

Changes in Chapter 5: Simplifying Formulas and Solving Equations

5A Simplifying Formulas and Like Terms

In the example of the constant term −4, the text said its number part is −7. It now says −4, because the number part of a constant is the constant itself.
The text said that 7a and 2b cannot be added. It now says they cannot be combined into a single term, because 7a plus 2b is a valid sum that, as the paragraph goes on to say, cannot be simplified further.
The text said we can only add or subtract terms with the same variable part. It now says that only such like terms can be combined into a single term by addition or subtraction, because unlike terms such as 7a and 2b can still be added as a sum.

5B Multiplying Terms with Associative and Distributive Properties

The reminder that we can only add or subtract like terms now says that only like terms can be combined into a single term by addition or subtraction, because unlike terms can still be added as a sum.
Near the start of the section, the text said many formulas have terms that cannot be added together, and it spoke of tripling the size of a rectangle. It now says unlike terms cannot be combined into a single term, and it speaks of tripling the length and width, because tripling both dimensions is what triples the perimeter to 3(2L plus 2W).
The text said unlike terms cannot be added or subtracted and that the key is to multiply the number parts. It now says unlike terms cannot be combined into a single term, and to multiply both the numerical coefficients and the variable parts, as in 7a times 3b equals 21ab.
The text said 2L and 2W cannot be added. It now says they cannot be combined into a single term, because 2L plus 2W is a valid sum, as the perimeter formula itself shows.
The text said the unlike terms in 9y minus 4 and −2y plus 8 cannot be added or subtracted. It now says they cannot be combined into a single term, since each expression is already a valid sum or difference.

5C Solving Equations with the Addition Property

The break-even point was described as where the company is not losing money and is starting to turn a profit. It now says the company is neither losing money nor earning a profit, because at the break-even point revenue equals cost, so the profit is exactly zero.
The text said that subtracting 19 is the same as adding −19 “from” both sides. It now says “to” both sides, because the same number is added to each side.

5E Steps to Solving General Linear Equations

The text said an equation with a finite number of solutions is conditional. It now says a finite, nonzero number, because zero is also a finite number, and an equation with no solution is a contradiction, not a conditional equation.
After 4x is subtracted from both sides of 5x plus 7 equals 4x minus 3, the text said the only x variables are on the “right” side. It now says the “left” side, because the result is x plus 7 equals −3.
Step 6 of the worked example that solves for c said we are multiplying w by −5. It now says c, the only variable in that example.
The third worked example printed its equation with plus 0.24 on the right side, but every step after it uses minus 0.24 and ends with the answer “All Real Numbers”; with plus 0.24 the equation would have no solution. It now prints minus 0.24, so the equation matches its worked solution.
In the first line of the third worked example's steps, one term was printed as 0.82x. It now reads 0.82m, matching the equation and every later step, which use only m.

5F Solving Proportions

The opening sentence said that “a special type of equation that has many applications are proportions”. It now says “is a proportion”, so the verb agrees with its singular subject.
The cross-product rule said that if c over d equals e over f, then c times f equals e times d. It now adds that d and f must be nonzero, because a fraction with a zero denominator is undefined.
The worked example said an equation that is not a proportion needs the methods of section 5D. It now points to the equation-solving methods of sections 5C to 5E, because such an equation may need the addition property and the general steps as well as the multiplication property.
The practice directions now also say to exclude any value that makes an original denominator zero and to check each answer in the original proportion. Several of these problems have a variable in a denominator, and a value that makes a denominator zero cannot be a solution.

5G Reading, Understanding, Solving and Graphing Inequalities

Edition note: Example 19 asks for −4p plus 1 to be at least −7 and at most −19 at the same time, which no number can do, because −19 is less than −7 (one part gives p at most 2, the other p at least 5). The example is kept exactly as printed, with an added note saying so, because the book does not show which bound was intended.
In the paragraph on less than, the text said the closed end points toward the smaller number for all of the inequality symbols. It now says the four order symbols (less than, less than or equal to, greater than, greater than or equal to), because the not-equal symbol, also taught here, does not point toward either number.
In the paragraph on greater than, the text again said the symbol points toward the smaller number for all of the inequality symbols. It now says the four order symbols, because the not-equal symbol does not point toward either number.
In the explanation of the graph of x greater than 3, “shades all the number to the right” now reads “shades all the numbers to the right”.
In the explanation of the graph of x less than or equal to 5, “shades all the number to the left” now reads “shades all the numbers to the left”.
A repeated word was removed: the greater than or equal to symbol was said to look like the “greater than than” symbol, and it now says the “greater than” symbol.
The practice directions said “Solve the following proportion problems”, the wording used for the proportions section, although these problems are about inequalities. They now say “Complete the following inequality problems” and add that problems 57 to 70 ask you to solve each inequality, since those problems have no directions of their own.
Problem 1: The answer said only “correctly”. It now also says that 17 is larger than 12, because the directions ask which number is larger as well as whether the symbol is used correctly.
Problem 3: The answer said “correctly”, but −7 is larger than −11, so the statement that −7 is less than −11 is false. The answer now says that −7 is larger than −11 and that the inequality is used incorrectly.
Problem 5: The answer said only “incorrectly”. It now also says that 1/4 equals 0.25, because the directions ask whether the numbers are equal or which is larger as well as whether the symbol is used correctly.
Problem 7: The answer said only “correctly”. It now also says that 7 5/6 is larger than 7, because the directions ask which number is larger as well as whether the symbol is used correctly.
Problem 9: The answer said only “incorrectly”. It now also says that the two values, both −21, are equal, because the directions ask whether the numbers are equal as well as whether the symbol is used correctly.
Problem 11: The answer said “correctly”, but −19 is less than 14, so the statement that −19 is greater than 14 is false. The answer now says that 14 is larger than −19 and that the inequality is used incorrectly.
Problem 13: The answer said only “correctly”. It now also says that 1/7 is larger than −3/4, because the directions ask which number is larger as well as whether the symbol is used correctly.
Problem 15: The answer said only “correctly”. It now also says that 7.6 is larger than 5.3, because the directions ask which number is larger as well as whether the symbol is used correctly.
Problem 17: The answer said only “correctly”. It now also puts the three numbers in order (−9 is less than 0, which is less than 14), because the directions ask which number is larger as well as whether the symbols are used correctly.
Problem 19: The answer said only “incorrectly”. It now also explains why: the first two values are both −23, so the strict less-than between them is false, while both are less than 0.
Problem 21: The answer said only “correctly”. It now also says that 3 is larger than −7, because the directions ask which number is larger as well as whether the symbol is used correctly.
After saying to switch the inequality from less than to greater than, the text printed the two resulting true statements still with less than (positive 30 less than −20, and positive 3 less than −2), which are false. They now read positive 30 greater than −20 and positive 3 greater than −2, and in the math each plus sign is now attached to its number instead of to the word statement.
Problem 31: The answer graph did not match the inequality x greater than 14 and less than 25: it was drawn on a number line from −2 to 7 and shaded only from between −1 and 0 up to 3. The answer is now a new number line from 13 to 26, shaded between 14 and 25 with a parenthesis at each end, because neither endpoint is included.
Problem 37: The answer graph did not match the inequality −3.25 less than or equal to x less than or equal to 1.75: it began with a parenthesis between −2 and −1 and ended with a bracket between 1 and 2. The answer is now a new number line from −6 to 4, shaded between brackets at −3.25 and 1.75, because both endpoints are included.
The graph of −4 less than or equal to x less than 2 had its endpoints the wrong way around, a parenthesis at −4 and a bracket at 2, although the text just above it says to use a bracket on −4 and a parenthesis on 2. The graph is redrawn with a bracket at −4 and a parenthesis at 2; the shaded stretch is unchanged.

Chapter 5 Review

The review said we cannot add 7f to 5w. It now says we cannot combine 7f and 5w into a single term, because unlike terms can still be added as a sum.
The review said a proportion is solved by setting the cross products equal and solving. It now adds that the original denominators must be nonzero and that the solution must be checked so it does not make either original denominator zero, because a fraction with a zero denominator is undefined.
Problem 22 printed 0.05a plus 1.9 equals 0.03x minus 0.4, mixing the variables a and x, although every other equation in this set has one variable. It now reads 0.03a, so the problem can be solved for a like its neighbors.
Problem 53: The answer was printed as 0.0000624 less than 0,05, with a comma in place of the decimal point, which a screen reader reads as 0 comma 05. The answer now reads “The P-value is less than 0.05”, followed by 0.0000624 less than 0.05.
Problem 60: This problem asks for 3x plus 2 to be greater than −1 and less than −7 at the same time, which no number can do, because −7 is less than −1 (one part gives x greater than −1, the other x less than −3). The problem is kept exactly as printed, with an added note saying so, and an answer is added that says it has no solution, because the book does not show which bound was intended.

Changes in Chapter 6: Equation Applications

Chapter 6: Introduction

The paragraph on uses of solving equations said that solving a regression equation shows how much of its product a company should make to minimize costs and maximize profits; it now says solving it shows how much is needed to reach a stated cost or profit target. Solving an equation finds the input that gives a chosen value, while finding the lowest cost or highest profit takes more analysis than solving one equation.

6B Bar Graphs, Pie Charts and Percent Applications

In the pie chart discussion, the text said the number of employees who prefer Kaiser is about 11% higher than the number who prefer Aetna; it now says Kaiser's share (41%) is about 11 percentage points higher than Aetna's (30%) and gives the exact shares from the counts. The original mixed up two comparisons: 33 employees is 37.5% more than 24, so 11 is the gap between the two shares, not a percent increase.
Problem 47: The answer said there were 30% more older adults than young adults; it now gives the difference as 58% minus 28%, which is 30 percentage points. Subtracting one share of the pie chart from another gives percentage points, not a percent increase, and the chart's counts show about twice as many older adults as young adults.
Problem 59: The answer said 10% more patients went to telemetry than to the ICU; it now gives the difference as 13% minus 3%, which is 10 percentage points. Subtracting one share of the pie chart from another gives percentage points, not a percent increase, and the chart's counts (407 telemetry, 87 intensive care) show more than four times as many.
Problem 65: The answer said 8% more students were married than divorced; it now gives the difference as 14% minus 6%, which is 8 percentage points. Subtracting one share of the pie chart from another gives percentage points, not a percent increase, and the chart's counts (2,341 married, 1,104 divorced) show about twice as many.
Problem 39: The problem asked how many times a slot machine that wins about 8% of the time must be played to win twice, and the answer said 25 times; it now asks how many plays give an expected total of two wins, and the answer says 25 plays give that expected total but do not guarantee two wins. A win rate of 8% means 25 plays produce 2 wins on average, while any actual run of 25 plays can produce fewer.
Problem 69: The problem applied a 57.5% turnout rate, measured among eligible voters, to all 199,182 people who live in Santa Clarita; it now asks the reader to suppose there are 199,182 eligible voters and to round to the nearest whole person. The answer is still 114,530 but now reads approximately 114,530 eligible voters, because a rate for eligible voters applies only to eligible voters and 57.5% of 199,182 is 114,529.65.
Problem 67: The problem asked for the total number of people in the apartment complex and the answer said 264; it now asks for the estimated total rounded to the nearest whole person, and the answer reads approximately 264 and gives the unrounded estimate. Since 29 divided by 0.11 is about 263.64, not a whole number, 264 is a rounded estimate.
Problem 71: The problem asked what percent of the doctor's patients are obese according to the sample, and the answer said 23.98%; it now asks about the doctor's sampled patients, rounded to the nearest hundredth of a percent, and the answer reads approximately 23.98%. The count of 59 out of 246 describes only the sample, and 59 divided by 246 is about 23.984%, so 23.98% is rounded.
Problem 68: The problem described a real Super Bowl as the most-watched game in NFL history, with 117.7 million viewers, in a year when 318.9 million people lived in the U.S.; it now asks the reader to suppose a televised game has 117.7 million viewers in a population of 318.9 million people. The numbers are the same, but they are now assumptions for the exercise rather than facts this edition has not verified.
Problem 70: The problem said a recent Gallup poll shows that 62% of Americans think the U.S. will be cashless in their lifetime; it now says to assume that 62% for this exercise. The numbers are unchanged, and the poll is no longer stated as a current fact, since this edition has not verified it.
Problem 37: The problem said it is estimated that about 5.3% of Americans over age 65 live in nursing homes; it now says to assume 5.3% for this exercise. The numbers and the answer of 365.7 people are unchanged, and the rate is no longer stated as a current fact, since this edition has not verified it.
Problem 40: The problem said it is estimated that 10.4% of children in California have some sort of disability; it now says to assume 10.4% for this exercise. The numbers are unchanged, and the rate is no longer stated as a current fact, since this edition has not verified it.
In the worked example on type 2 diabetes, the text said it is estimated that about 39% of American adults develop the disease in their lifetime and then applied that rate to all 3500 people in a hometown; it now says to assume 39% and counts 3500 adults. A rate measured among adults applies only to adults, and the 39% is now an assumption because this edition has not verified it as a current medical fact.
The conclusion of the type 2 diabetes example said about 1365 people in the hometown can be expected to develop the disease; it now says 1365 adults, to match the example's count of 3500 adults (39% of 3500 is 1365).
The rule for solving a proportion by setting the cross products equal now begins "With the original denominators nonzero". Cross products give a correct answer only when the original fractions are defined, and a fraction with a zero denominator is undefined.

6C Commission, Interest, Tax, Markup and Discount

The explanation of simple interest said "It a big part" and now says "It is a big part", and r is now called the annual interest rate. The formula measures the time t in years, so the rate must be a yearly rate.
The explanation of sales tax said you need to pay taxes whether you buy coffee or a car and gave 9.25% and 8.5% as tax rates in parts of California; it now asks how sales taxes work, says to assume the stated item is taxable, and calls 9.25% and 8.5% example rates for the exercises. Not every purchase is taxed and local rates change, so the rates are now assumptions for the exercises rather than claims about current taxes.
The reminder before the practice problems said "percentages rates" and now says "percentage rates". This is a grammar fix.
Problem 15: The question said "the stores markup rate" and now says "the store's markup rate". This adds a missing apostrophe; nothing else changed.
Problem 20: The question asked for the regular price of the shed, although the problem is about a book bought on sale; it now asks for the regular price of the book.

6D Classic Algebraic Problem Solving

Problem 5: The answer said 19; it is now 24. Half of 24 is 12, and 12 plus 7 is 19 as the problem states, while half of 19 plus 7 is only 16.5.
Problem 9: The problem said "the number subtracted by 71" and now says "the number decreased by 71", which plainly means the number minus 71. With that reading, one sixth of 90 plus 4 is 19 and 90 minus 71 is 19, so the answer of 90 is unchanged.
Problem 12: The problem said "She as a total of 78 flowers" and now says "She has a total of 78 flowers". This is a typing fix; the answer is unchanged.
The opening paragraph said algebra can be "a useful tool to solving" and now says "a useful tool for solving". This is a grammar fix.
Problem 21: The problem said the ancient Greeks were famous for using the golden rectangle in their architecture and that a wall of a temple in Greece has that shape; it now keeps the same definition (length about 1.618 times the width) and asks the reader to suppose a rectangular wall has those proportions. The numbers and the answer (length 80.9 feet, width 50 feet) are unchanged, and the historical claim, which this edition has not verified, was removed.

Chapter 6 Review

Problem 35: The answer said 7,188 full time and 3,620 part time students; it now says 7,188 part time and 3,620 full time. The problem says part time is 52 less than twice full time, and twice 3,620 minus 52 is 7,188, so the labels in the original answer were swapped.
Problem 41: The answer gave 19.29% next to the fraction 98 over 508; it now says approximately 19.29%, found as 100 times 98 divided by 508, using the estimated bar height. The fraction itself is about 0.1929, so it must be multiplied by 100 to give a percent, and the count of 98 is read from the bar graph, so the result is approximate.
The reminder about classic algebra problems said "To solve a classic algebra problems" and now says "To solve classic algebra problems". This is a grammar fix.
Problem 24: The problem said it is estimated that about 5.3% of Americans over age 65 live in nursing homes; it now says to assume 5.3% for this exercise, and "a nursing homes" now reads "nursing homes". The numbers are unchanged, and the rate is no longer stated as a current fact, since this edition has not verified it.
Problem 32: The question said "the stores markup rate" and now says "the store's markup rate". This adds a missing apostrophe; nothing else changed.
The sentence introducing the political party bar graph said students were asked "what their political party they affiliate with"; it now says "which political party they affiliate with". This is a grammar fix; the survey and its 508 responses are unchanged.

Changes in Chapter 7: Equation of a Line, Slope, and the Rectangular Coordinate System

7A Rectangular Coordinate System and Scatterplots

Edition note: Added to point out that the text names the y-axis point (0, −5) while the graph labels it (0, −6). Both points have x-coordinate 0 and lie on the y-axis, so the point is still a y-intercept either way; the text and graph are kept as printed.
Problem 3: The question now asks which points are x-intercepts and which are y-intercepts, instead of asking for one point each. The point (0, 0) lies on both axes, so this problem has two x-intercepts, (−2, 0) and (0, 0), as the printed answer already shows.
Problem 7: The question now asks which points are x-intercepts and which are y-intercepts, instead of asking for one point each. The point (0, 0) lies on both axes, so this problem has two x-intercepts, (−3, 0) and (0, 0), as the printed answer already shows.
Problem 11: The question now asks which points are x-intercepts and which are y-intercepts, instead of asking for one point each. The point (0, 0) lies on both axes, so this problem has two x-intercepts, (−2.5, 0) and (0, 0), as the printed answer already shows.

7B Slope of a Line and Average Rates of Change

A typo was fixed in the question that introduces the definition of slope: "What it slope?" now reads "What is slope?"
A typo was fixed in the explanation of horizontal change: "your can subtract" now reads "you can subtract".
The rule for parallel lines now says that nonvertical parallel lines have the same slope, and that vertical lines are parallel to each other with undefined slope. The original said all parallel lines have the same slope, but a vertical line has no numerical slope, as the section explains earlier.
The rule for perpendicular lines now says that perpendicular lines with nonzero slopes have slopes that are negative reciprocals of each other, and adds that a horizontal line and a vertical line are also perpendicular. A horizontal line has slope 0, which has no reciprocal, and a vertical line has undefined slope, so the original rule ("the opposite or negative reciprocal") could not cover that pair.
The opening IBM example now gives the two prices, $186.91 at the end of September 2014 and $160.44 at the end of December, as assumptions for the exercise, and drops the statement that the price rose and fell in between. The numbers are unchanged; the example uses only these two prices, so it does not depend on the stock's actual history.
Where the section returns to the IBM example, the two prices are again stated as assumptions for the exercise, and the statement that the price rose and fell in between is removed, to match the opening. The prices and dates are unchanged.
The explanation of the slope now says the stock price fell by about $8.82 per month on average over the three months, not that it was falling every month. Two end prices give only an average rate, and −26.47 ÷ 3 is about −8.8233, so $8.82 is a rounded value.

7C Finding the Equation of a Line

The slope-intercept equation y = mx + b is now described as the equation of a nonvertical line. A vertical line has undefined slope and is written as x equals a constant, as shown later in the section.
The 2009 and 2013 national debt and unemployment figures (11.9 trillion dollars and 9.9 percent; 16.7 trillion dollars and 6.7 percent) are now stated as assumptions for the exercise instead of as historical facts. The numbers are unchanged; the text does not say which official measures they come from, so they are treated as given values.
Problem 42: The national debt and unemployment figures in this problem are now stated as assumptions for the exercise, matching the change earlier in the section. The numbers, ordered pairs and questions are unchanged.
The reminder of the IBM example now says the two prices were assumed for the exercise and drops the statement that the price rose and fell in between, matching the earlier IBM example. The prices and the slope of about −$8.82 per month are unchanged.
In the IBM slope calculation, the equals sign before −8.823 is now an approximately equal sign. (160.44 − 186.91) ÷ (12 − 9) = −26.47 ÷ 3 = −8.8233 with the 3 repeating, so −8.823 is a rounded value, not an exact one.
The sentence after the slope now says that, under the exercise assumptions, the stock price decreased by about $8.82 per month on average over the three months. The original said the price is decreasing at a rate of $8.82 per month, but two end prices give only an average rate.
The step that finds the y-intercept now says it uses the slope rounded to three decimal places, −8.823, and so estimates the intercept. The calculation and its result, about $266.32, are unchanged.
The meaning of the y-intercept now says the line extrapolates a price of about $266.32 at month zero, which is outside the three months of data and is not an observed price. The original called it the predicted price of the stock in month zero.
The equation y = −8.82x + 266.32 is now called an approximate model built from the rounded slope and intercept, not the equation of the line that describes the stock price. With these rounded values the line does not pass exactly through both given points; the prediction of $142.84 for month 14 is kept, while the unrounded line gives about $142.79.
Problem 37: A typo was fixed: "the companies costs" now reads "the company’s costs".

7D Systems of Linear Equations

The break-even example now compares a cost equation with a revenue equation and defines the break-even point as where cost and revenue are equal. The original said cost and profit, but break-even means revenue equals cost, which is where profit is zero.
The second of the three cases now says the lines are distinct parallel lines that do not meet. This keeps it separate from the third case, where both equations describe the same line and every point on it is a solution.
The description of a system with no solution now says the two lines are distinct parallel lines. Two equations for the same line also have equal slopes, but that system has infinitely many solutions, not none.
The graph showing two lines that are the same line was redrawn on a numbered grid, and its description was rewritten to match. The original sketch had no numbers, so the new graph is an illustration, the line y = −x/3 through the marked points (0, 0) and (3, −1), not a copy of the original line.
The graph for the example with 5x + 2y = 10 and y = −(5/2)x + 5 was redrawn on a numbered grid as one line through the marked points (0, 5) and (3, −5/2), crossing the x-axis at (2, 0), and its description now names that line. The original drawing had no numbers on its grid and stopped at the y-axis.

Chapter 7 Review

Problem 47: The key printed (0, 4); it is now (0, −4). Both equations, y = 7x − 4 and y = −(2/3)x − 4, have y-intercept −4, so the lines meet at (0, −4).
Problem 1: The key named only the x-intercept and the y-intercept; it now also gives the quadrant of each of the other three points, as the problem asks: (−5, −2) in Quadrant III, (−4, 2) in Quadrant II and (1, −3) in Quadrant IV.
Problem 3: The key named only the x-intercept and the y-intercept; it now also gives the quadrant of each of the other three points, as the problem asks: (1.5, 5.5) in Quadrant I, (−1.25, −3.25) in Quadrant III and (−1.5, 3.25) in Quadrant II.
Problem 5: The problem now states that moving 3 units to the right along the line changes y by −8 units, and asks for the slope from that change. The graph has no numbers on its axes, so the change is now given in words rather than left to counting grid squares; it matches the printed answer, m = −8/3, which is unchanged.
Problem 27: The problem now states that moving 1 unit to the right along the line changes y by −4 units, and asks for the equation from that slope and the labeled point (0, 0). The graph has no numbers on its axes, so the change is now given in words rather than left to counting grid squares; it matches the printed answer, y = −4x, which is unchanged.
Problem 26: The problem now states in words that moving 4 units to the right along the line changes y by 1 unit. The graph has no numbers on its axes and labels only the y-intercept, (0, 3), so the slope could be found only by counting grid squares, which a reader who cannot see the graph cannot do; the task, to find the equation of the line in slope-intercept form, is unchanged.
Problem 28: The problem now states in words that moving 5 units to the right along the line changes y by −2 units. The graph has no numbers on its axes and labels only the y-intercept, (0, −3), so the slope could be found only by counting grid squares, which a reader who cannot see the graph cannot do; the task, to find the equation of the line in slope-intercept form, is unchanged.
The chapter summary now says that any two distinct points give the exact slope and y-intercept only when the points lie exactly on a nonvertical line; when a scatterplot only roughly follows a line, a line through two points is an approximation. The misspelling "y-intecept" was also corrected.
A typo was fixed in the directions for the substitution problems: "Solve each the following" now reads "Solve each of the following".
Problem 43: A typo was fixed: "the companies average profits" now reads "the company’s average profits".
Problem 44: A typo was fixed in the last question: "predict the how much" now reads "predict how much".