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== 3.1#h(0.6em)Use a Problem-Solving Strategy

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Translate “6 less than twice #emph[x]” into an algebraic expression. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2 x minus 6")[$2 x − 6$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 2 over 3 x equals 24 .")[$frac(2, 3) x = 24 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x equals 36")[$x = 36$]
]

Solve: #math.equation(block: false, alt: "3 x plus 8 equals 14 .")[$3 x + 8 = 14 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x equals 2")[$x = 2$]
]
]


=== Approach Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford

The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?

How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?

#figure(figph[A student is shown with thought bubbles saying “I don’t know whether to add, subtract, multiply, or divide!,” “I don’t understand word problems!,” “My teachers never explained this!,” “If I just skip all the word problems, I can probably still pass the class,” and “I just can’t do this!”], alt: "A student is shown with thought bubbles saying “I don’t know whether to add, subtract, multiply, or divide!,” “I don’t understand word problems!,” “My teachers never explained this!,” “If I just skip all the word problems, I can probably still pass the class,” and “I just can’t do this!”", caption: [Negative thoughts can be barriers to success.])

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in  and say them out loud.

#figure(figph[A student is shown with thought bubbles saying “While word problems were hard in the past, I think I can try them now,” “I am better prepared now. I think I will begin to understand word problems,” “I think I can! I think I can!,” and “It may take time, but I can begin to solve word problems.”], alt: "A student is shown with thought bubbles saying “While word problems were hard in the past, I think I can try them now,” “I am better prepared now. I think I will begin to understand word problems,” “I think I can! I think I can!,” and “It may take time, but I can begin to solve word problems.”", caption: [Thinking positive thoughts is a first step towards success.])

Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!


=== Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.

Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Use a Problem-Solving Strategy to Solve Word Problems.]

+ #strong[Read] the problem. Make sure all the words and ideas are understood.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for. Choose a variable to represent that quantity.
+ #strong[Translate] into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.
]

#examplebox("Example 1")[][
Pilar bought a purse on sale for \$18, which is one-half of the original price. What was the original price of the purse?

#solutionbox[
#strong[Step 1. Read] the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.

- #emph[In this problem, is it clear what is being discussed? Is every word familiar?]

#strong[Step 2. Identify] what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

- #emph[In this problem, the words “what was the original price of the purse” tell us what we need to find.]

#strong[Step 3. Name] what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.

- Let #math.equation(block: false, alt: "p equals")[$p =$] the original price of the purse.

#strong[Step 4. Translate] into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.

Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Restate the problem in one sentence with all the important information.], [#figure(figph[The image shows the equation '18 is one-half the original price.', with brackets indicating the different parts of the phrase to denote algebraic terms.], alt: "The image shows the equation '18 is one-half the original price.', with brackets indicating the different parts of the phrase to denote algebraic terms.", caption: none)]),
  [Translate into an equation.], [#figure(figph[A mathematical equation displaying 18 = 1/2 \* p, which can be solved for the variable p.], alt: "A mathematical equation displaying 18 = 1/2 * p, which can be solved for the variable p.", caption: none)],
))

#strong[Step 5. Solve] the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Solve the equation.], [#figure(figph[A mathematical equation is displayed, showing '18 = 1/2 p' against a plain white background.], alt: "A mathematical equation is displayed, showing '18 = 1/2 p' against a plain white background.", caption: none)]),
  [Multiply both sides by 2.], [#figure(figph[A mathematical equation is displayed, showing the expression 2 multiplied by 18, which is set equal to 2 multiplied by one-half p, represented as 2 \* 18 = 2 \* (1/2)p.], alt: "A mathematical equation is displayed, showing the expression 2 multiplied by 18, which is set equal to 2 multiplied by one-half p, represented as 2 * 18 = 2 * (1/2)p.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the mathematical equation '36 = p' presented in a clean, straightforward manner on a white background.], alt: "The image shows the mathematical equation '36 = p' presented in a clean, straightforward manner on a white background.", caption: none)],
))

#strong[Step 6. Check] the answer in the problem to make sure it makes sense. We solved the equation and found that #math.equation(block: false, alt: "p equals 36 ,")[$p = 36 ,$] which means “the original price” was \$36.

- #emph[Does \$36 make sense in the problem? Yes, because 18 is one-half of 36, and the purse was on sale at half the original price.]

#strong[Step 7. Answer] the question with a complete sentence. The problem asked “What was the original price of the purse?”

- #emph[The answer to the question is: “The original price of the purse was \$36.”] #linebreak()

If this were a homework exercise, our work might look like this:

Pilar bought a purse on sale for \$18, which is one-half the original price. What was the original price of the purse?

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [Let #math.equation(block: false, alt: "p equals")[$p =$] the original price.]),
  [], [18 is one-half the original price.],
  [], [#figure(figph[A mathematical equation is displayed on a white background, reading 18 = 1/2p, where 18 is equal to one-half multiplied by the variable p.], alt: "A mathematical equation is displayed on a white background, reading 18 = 1/2p, where 18 is equal to one-half multiplied by the variable p.", caption: none)],
  [Multiply both sides by 2.], [#figure(figph[A mathematical equation showing the step of multiplying both sides by 2: '2 times 18 equals 2 times one-half p'. The number 2 is highlighted in red on both sides.], alt: "A mathematical equation showing the step of multiplying both sides by 2: '2 times 18 equals 2 times one-half p'. The number 2 is highlighted in red on both sides.", caption: none)],
  [Simplify.], [#figure(figph[An equation showing 36 equals p.], alt: "An equation showing 36 equals p.", caption: none)],
  [Check. Is \$36 a reasonable price for a purse?], [],
  [Yes.], [],
  [Is 18 one half of 36?], [],
  [#math.equation(block: false, alt: "18 equals ? the fraction 1 over 2 times 36")[$18 limits(=)^(?) frac(1, 2) ⋅ 36$]], [],
  [#math.equation(block: false, alt: "18 equals 18 ✓")[$18 = 18 ✓$]], [],
  [], [The original price of the purse was \$36.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Joaquin bought a bookcase on sale for \$120, which was two-thirds of the original price. What was the original price of the bookcase?

#solutionbox[
\$180
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Two-fifths of the songs in Mariel’s playlist are country. If there are 16 country songs, what is the total number of songs in the playlist?

#solutionbox[
40
]
]

Let’s try this approach with another example.

#examplebox("Example 2")[][
Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what we are looking for.], [How many boys were in the study group?],
  [#emph[Step 3. Name.] Choose a variable to represent the number of boys.], [Let #math.equation(block: false, alt: "n equals")[$n =$] the number of boys.],
  [#emph[Step 4. Translate.] Restate the problem in one sentence with all the important information.], [#figure(figph[A word problem is shown, breaking down the sentence 'The number of girls (11) was three more than twice the number of boys' into its constituent mathematical components.], alt: "A word problem is shown, breaking down the sentence 'The number of girls (11) was three more than twice the number of boys' into its constituent mathematical components.", caption: none)],
  [Translate into an equation.], [#figure(figph[A mathematical equation is displayed with the numbers and variable '11 = 2b + 3' in a horizontal line, on a plain white background.], alt: "A mathematical equation is displayed with the numbers and variable '11 = 2b + 3' in a horizontal line, on a plain white background.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [#figure(figph[A clear image displaying the mathematical equation '11 = 2b + 3' centered on a plain white background.], alt: "A clear image displaying the mathematical equation '11 = 2b + 3' centered on a plain white background.", caption: none)],
  [Subtract 3 from each side.], [#figure(figph[A mathematical equation, 11 - 3 = 2b + 3 - 3, is shown on a white background, with the '-3' terms highlighted in red on both sides of the equality.], alt: "A mathematical equation, 11 - 3 = 2b + 3 - 3, is shown on a white background, with the '-3' terms highlighted in red on both sides of the equality.", caption: none)],
  [Simplify.], [#figure(figph[A simple algebraic equation is displayed, showing '8 = 2b' on a white background, representing a common problem in mathematics where one must solve for the variable 'b'.], alt: "A simple algebraic equation is displayed, showing '8 = 2b' on a white background, representing a common problem in mathematics where one must solve for the variable 'b'.", caption: none)],
  [Divide each side by 2.], [#figure(figph[A mathematical equation shows '8/2 = 2b/2', with the number 2 in the denominator highlighted in red on both sides, indicating a step in solving for 'b'.], alt: "A mathematical equation shows '8/2 = 2b/2', with the number 2 in the denominator highlighted in red on both sides, indicating a step in solving for 'b'.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation '4 = b' is displayed prominently in the center of a plain white background.], alt: "The mathematical equation '4 = b' is displayed prominently in the center of a plain white background.", caption: none)],
  [#emph[Step 6. Check.] First, is our answer reasonable? Yes, having 4 boys in a study group seems OK. The problem says the number of girls was 3 more than twice the number of boys. If there are four boys, does that make eleven girls? Twice 4 boys is 8. Three more than 8 is 11.], [],
  [#emph[Step 7. Answer] the question.], [There were 4 boys in the study group.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 more than twice the number of notebooks. He bought 7 textbooks. How many notebooks did he buy?

#solutionbox[
2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?

#solutionbox[
7
]
]


=== Solve Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

#examplebox("Example 3")[][
The difference of a number and six is 13. Find the number.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem. Are all the words familiar?], []),
  [#emph[Step 2. Identify] what we are looking for.], [the number],
  [#emph[Step 3. Name.] Choose a variable to represent the number.], [Let #math.equation(block: false, alt: "n equals")[$n =$] the number.],
  [#emph[Step 4. Translate.] Remember to look for clue words like "difference... of... and..."], [],
  [Restate the problem as one sentence.], [#figure(figph[A mathematical word problem states: The difference of the number and 6 is 13. Light blue brackets underline 'The difference of the number and 6', 'is', and '13' to segment the sentence.], alt: "A mathematical word problem states: The difference of the number and 6 is 13. Light blue brackets underline 'The difference of the number and 6', 'is', and '13' to segment the sentence.", caption: none)],
  [Translate into an equation.], [#figure(figph[A simple algebraic equation is displayed on a white background, which reads 'n - 6 = 13'.], alt: "A simple algebraic equation is displayed on a white background, which reads 'n - 6 = 13'.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [#figure(figph[A mathematical equation is displayed on a white background, reading 'n - 6 = 13' in black text.], alt: "A mathematical equation is displayed on a white background, reading 'n - 6 = 13' in black text.", caption: none)],
  [Simplify.], [#figure(figph[The number 19 is shown in the top right corner of a white background, preceded by 'n =', indicating a count or sample size.], alt: "The number 19 is shown in the top right corner of a white background, preceded by 'n =', indicating a count or sample size.", caption: none)],
  [#emph[Step 6. Check.]], [],
  [The difference of 19 and 6 is 13. It checks!], [],
  [#emph[Step 7. Answer] the question.], [The number is 19.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The difference of a number and eight is 17. Find the number.

#solutionbox[
25
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The difference of a number and eleven is #math.equation(block: false, alt: "−7 .")[$−7 .$] Find the number.

#solutionbox[
4
]
]

#examplebox("Example 4")[][
The sum of twice a number and seven is 15. Find the number.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what we are looking for.], [the number],
  [#emph[Step 3. Name.] Choose a variable to represent the number.], [Let #math.equation(block: false, alt: "n equals")[$n =$] the number.],
  [#emph[Step 4. Translate.]], [],
  [Restate the problem as one sentence.], [#figure(figph[A mathematical statement reads: 'The sum of twice a number and 7 is 15'. Brackets below the text segment 'The sum of twice a number and 7' and below 'is' and '15' highlight their roles in forming an equation.], alt: "A mathematical statement reads: 'The sum of twice a number and 7 is 15'. Brackets below the text segment 'The sum of twice a number and 7' and below 'is' and '15' highlight their roles in forming an equation.", caption: none)],
  [Translate into an equation.], [#figure(figph[A mathematical equation is displayed with the expression], alt: "A mathematical equation is displayed with the expression", caption: none)],
  [#emph[Step 5. Solve] the equation.], [#figure(figph[The image displays the algebraic equation '2n + 7 = 15' centered on a white background, representing a linear equation in one variable.], alt: "The image displays the algebraic equation '2n + 7 = 15' centered on a white background, representing a linear equation in one variable.", caption: none)],
  [Subtract 7 from each side and simplify.], [#figure(figph[A simple algebraic equation is displayed, reading '2n = 8' against a plain white background, indicating a basic problem to solve for the variable 'n'.], alt: "A simple algebraic equation is displayed, reading '2n = 8' against a plain white background, indicating a basic problem to solve for the variable 'n'.", caption: none)],
  [Divide each side by 2 and simplify.], [#figure(figph[The image features the text 'n = 4' in the upper right corner against a plain white background.], alt: "The image features the text 'n = 4' in the upper right corner against a plain white background.", caption: none)],
  [#emph[Step 6. Check.]], [],
  [Is the sum of twice 4 and 7 equal to 15?], [],
  [#math.equation(block: false, alt: "2 times 4 plus 7, ≟, 15; 15, equals, 15 ✓")[$2 ⋅ 4 + 7 & ≟ & 15 \ 15 & = & 15 ✓$]], [],
  [#emph[Step 7. Answer] the question.], [The number is 4.],
))

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of four times a number and two is 14. Find the number.

#solutionbox[
3
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of three times a number and seven is 25. Find the number.

#solutionbox[
6
]
]

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

#examplebox("Example 5")[][
One number is five more than another. The sum of the numbers is 21. Find the numbers.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [We are looking for two numbers.],
  [#emph[Step 3. Name.] We have two numbers to name and need a name for each.], [], [],
  [Choose a variable to represent the first number.], [], [Let #math.equation(block: false, alt: "n equals 1 to the power st")[$n = 1^("st")$] number.],
  [What do we know about the second number?], [], [One number is five more than another.],
  [], [], [#math.equation(block: false, alt: "n plus 5 equals 2 to the power nd")[$n + 5 = 2^("nd")$] number],
  [#emph[Step 4. Translate.]Restate the problem as one sentence with all the important information.], [], [The sum of the 1#super[st] number and the 2#super[nd] number is 21.],
  [Translate into an equation.], [], [#figure(figph[An equation displays '1st number + 2nd number = 21', with light blue brackets segmenting each component of the expression.], alt: "An equation displays '1st number + 2nd number = 21', with light blue brackets segmenting each component of the expression.", caption: none)],
  [Substitute the variable expressions.], [], [#figure(figph[A mathematical equation is displayed on a white background, which reads 'n + n + 5 = 21'.], alt: "A mathematical equation is displayed on a white background, which reads 'n + n + 5 = 21'.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[A mathematical equation is displayed on a white background: n + n + 5 = 21. The variables and numbers are in black text, solving for 'n'.], alt: "A mathematical equation is displayed on a white background: n + n + 5 = 21. The variables and numbers are in black text, solving for 'n'.", caption: none)],
  [Combine like terms.], [], [#figure(figph[A mathematical equation '2n + 5 = 21' is displayed on a white background.], alt: "A mathematical equation '2n + 5 = 21' is displayed on a white background.", caption: none)],
  [Subtract 5 from both sides and simplify.], [], [#figure(figph[A simple algebraic equation is displayed on a white background, showing '2n = 16' in a dark, legible font. This equation can be solved by dividing both sides by 2, which yields n=8.], alt: "A simple algebraic equation is displayed on a white background, showing '2n = 16' in a dark, legible font. This equation can be solved by dividing both sides by 2, which yields n=8.", caption: none)],
  [Divide by 2 and simplify.], [], [#figure(figph[The text 'n = 8 1st number' is displayed on a white background, suggesting a mathematical or sequential enumeration context.], alt: "The text 'n = 8 1st number' is displayed on a white background, suggesting a mathematical or sequential enumeration context.", caption: none)],
  [Find the second number, too.], [], [#figure(figph[The image shows the mathematical expression 'n + 5' followed by the text '2nd number', suggesting a mathematical sequence or a problem's second term.], alt: "The image shows the mathematical expression 'n + 5' followed by the text '2nd number', suggesting a mathematical sequence or a problem's second term.", caption: none)],
  [], [], [#figure(figph[The image displays the arithmetic expression '8 + 5' with the number 8 in red and the plus sign and number 5 in black, all against a plain white background.], alt: "The image displays the arithmetic expression '8 + 5' with the number 8 in red and the plus sign and number 5 in black, all against a plain white background.", caption: none)],
  [], [], [#figure(figph[The number 13 is prominently displayed in the center of a plain white background.], alt: "The number 13 is prominently displayed in the center of a plain white background.", caption: none)],
  [#emph[Step 6. Check.]], [], [],
  [Do these numbers check in the problem?], [], [],
  [Is one number 5 more than the other?], [#math.equation(block: false, alt: "13 ≟ 8 plus 5")[$#h(1.6em) 13 ≟ 8 + 5$]], [],
  [Is thirteen 5 more than 8? Yes.], [#math.equation(block: false, alt: "13 equals 13 ✓")[$#h(1.6em) 13 = 13 ✓$]], [],
  [Is the sum of the two numbers 21?], [#math.equation(block: false, alt: "8 plus 13 ≟ 21")[$8 + 13 ≟ 21$]], [],
  [], [#math.equation(block: false, alt: "21 equals 21 ✓")[$#h(1.6em) 21 = 21 ✓$]], [],
  [#emph[Step 7. Answer] the question.], [], [The numbers are 8 and 13.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

#solutionbox[
9, 15
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

#solutionbox[
27, 31
]
]

#examplebox("Example 6")[][
The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [We are looking for two numbers.],
  [#emph[Step 3. Name.]], [], [],
  [Choose a variable.], [], [Let #math.equation(block: false, alt: "n equals 1 to the power st")[$n = 1^("st")$] number.],
  [One number is 4 less than the other.], [], [#math.equation(block: false, alt: "n minus 4 equals 2 to the power nd")[$n − 4 = 2^("nd")$] number],
  [#emph[Step 4. Translate.]], [], [],
  [Write as one sentence.], [], [The sum of the 2 numbers is negative 14.],
  [Translate into an equation.], [], [#figure(figph[An image displaying a mathematical statement: '1st number + 2nd number is negative fourteen', with individual parts underlined in light blue brackets.], alt: "An image displaying a mathematical statement: '1st number + 2nd number is negative fourteen', with individual parts underlined in light blue brackets.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[The image shows the algebraic equation n + n - 4 = -14.], alt: "The image shows the algebraic equation n + n - 4 = -14.", caption: none)],
  [Combine like terms.], [], [#figure(figph[A mathematical equation is displayed against a white background: n + n - 4 = -14.], alt: "A mathematical equation is displayed against a white background: n + n - 4 = -14.", caption: none)],
  [Add 4 to each side and simplify.], [], [#figure(figph[A mathematical equation is displayed against a white background, which reads '2n - 4 = -14'.], alt: "A mathematical equation is displayed against a white background, which reads '2n - 4 = -14'.", caption: none)],
  [Simplify.], [], [#figure(figph[A mathematical equation reads '2n = -10' presented on a plain white background, likely a problem from algebra or basic math.], alt: "A mathematical equation reads '2n = -10' presented on a plain white background, likely a problem from algebra or basic math.", caption: none)],
  [], [], [#figure(figph[The image shows mathematical notation and text against a white background. It displays 'n = -.5 1\*number'.], alt: "The image shows mathematical notation and text against a white background. It displays 'n = -.5 1*number'.", caption: none)],
  [], [], [#figure(figph[An algebraic expression 'n - 4' is displayed next to the text '2nd number' on a white background.], alt: "An algebraic expression 'n - 4' is displayed next to the text '2nd number' on a white background.", caption: none)],
  [], [], [#figure(figph[The image displays a mathematical expression '-5-4' in red and black numerals on a white background, representing a simple subtraction problem.], alt: "The image displays a mathematical expression '-5-4' in red and black numerals on a white background, representing a simple subtraction problem.", caption: none)],
  [], [], [#figure(figph[The number -9 is displayed centered on a white background.], alt: "The number -9 is displayed centered on a white background.", caption: none)],
  [#emph[Step 6. Check.]], [], [],
  [Is −9 four less than −5?], [#math.equation(block: false, alt: "−5 minus 4 ≟ −9")[$#h(1.1em) −5 − 4 ≟ −9$]], [],
  [], [#math.equation(block: false, alt: "−9 equals −9 ✓")[$#h(2.7em) −9 = −9 ✓$]], [],
  [Is their sum −14?], [#math.equation(block: false, alt: "−5 plus open parenthesis −9 close parenthesis ≟ −14")[$−5 + ( −9 ) ≟ −14$]], [],
  [], [#math.equation(block: false, alt: "−14 equals −14 ✓")[$#h(2.3em) −14 = −14 ✓$]], [],
  [#emph[Step 7. Answer] the question.], [], [The numbers are −5 and −9.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−15 , −8")[$−15 , −8$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two numbers is #math.equation(block: false, alt: "−18 .")[$−18 .$] One number is 40 more than the other. Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−29 , 11")[$−29 , 11$]
]
]

#examplebox("Example 7")[][
One number is ten more than twice another. Their sum is one. Find the numbers.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what you are looking for.], [], [We are looking for two numbers.],
  [#emph[Step 3. Name.]], [], [],
  [Choose a variable.], [], [Let #math.equation(block: false, alt: "x equals 1 to the power st")[$x = 1^("st")$] number.],
  [One number is 10 more than twice another.], [], [#math.equation(block: false, alt: "2 x plus 10 equals 2 to the power nd")[$2 x + 10 = 2^("nd")$] number],
  [#emph[Step 4. Translate.]], [], [],
  [Restate as one sentence.], [], [Their sum is one.],
  [], [], [The sum of the two numbers is 1.],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation is displayed, reading 'x + 2x + 10 = 1'. The equation shows variables and constants combined with addition and an equality sign.], alt: "A mathematical equation is displayed, reading 'x + 2x + 10 = 1'. The equation shows variables and constants combined with addition and an equality sign.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [],
  [Combine like terms.], [], [#figure(figph[A mathematical equation is displayed, showing 'x + 2x + 10 = 1' in a horizontal layout on a white background.], alt: "A mathematical equation is displayed, showing 'x + 2x + 10 = 1' in a horizontal layout on a white background.", caption: none)],
  [Subtract 10 from each side.], [], [#figure(figph[The image displays a mathematical equation: '3x + 10 = 1'. This is an algebraic linear equation where 'x' is the unknown variable, shown in a clean, straightforward white background.], alt: "The image displays a mathematical equation: '3x + 10 = 1'. This is an algebraic linear equation where 'x' is the unknown variable, shown in a clean, straightforward white background.", caption: none)],
  [Divide each side by 3.], [], [#figure(figph[A simple algebraic equation is displayed, stating '3x = -9' in a clear, dark font against a plain white background.], alt: "A simple algebraic equation is displayed, stating '3x = -9' in a clear, dark font against a plain white background.", caption: none)],
  [], [], [#figure(figph[The image displays mathematical text with 'x = -3' and '1st number' written on a white background, suggesting an equation or a label for a numerical value.], alt: "The image displays mathematical text with 'x = -3' and '1st number' written on a white background, suggesting an equation or a label for a numerical value.", caption: none)],
  [], [], [#figure(figph[The image displays the algebraic expression '2x + 10' followed by '2nd number' in a simple, clear font on a white background.], alt: "The image displays the algebraic expression '2x + 10' followed by '2nd number' in a simple, clear font on a white background.", caption: none)],
  [], [], [#figure(figph[A mathematical expression '2(-3) + 10' is shown in black text on a white background, representing a calculation involving multiplication and addition with a negative number.], alt: "A mathematical expression '2(-3) + 10' is shown in black text on a white background, representing a calculation involving multiplication and addition with a negative number.", caption: none)],
  [], [], [#figure(figph[The number 4 is displayed.], alt: "The number 4 is displayed.", caption: none)],
  [#emph[Step 6. Check.]], [], [],
  [Is ten more than twice −3 equal to 4?], [#math.equation(block: false, alt: "2 open parenthesis −3 close parenthesis plus 10 ≟ 4")[$2 ( −3 ) + 10 ≟ 4$]], [],
  [], [#math.equation(block: false, alt: "−6 plus 10 ≟ 4")[$#h(1em) −6 + 10 ≟ 4$]], [],
  [], [#math.equation(block: false, alt: "4 equals 4 ✓")[$#h(3.8em) 4 = 4 ✓$]], [],
  [Is their sum 1?], [#math.equation(block: false, alt: "−3 plus 4 ≟ 1")[$#h(1.5em) −3 + 4 ≟ 1$]], [],
  [], [#math.equation(block: false, alt: "1 equals 1 ✓")[$#h(3.8em) 1 = 1 ✓$]], [],
  [#emph[Step 7. Answer] the question.], [], [The numbers are −3 and 4.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One number is eight more than twice another. Their sum is negative four. Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−4 , 0")[$−4 , 0$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One number is three more than three times another. Their sum is #math.equation(block: false, alt: "−5 .")[$−5 .$] Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−3 , −2")[$−3 , −2$]
]
]

Some number problems involve consecutive integers. #emph[Consecutive integers] are integers that immediately follow each other. Examples of consecutive integers are:

#math.equation(block: true, alt: "1 , 2 , 3 , 4; minus 10 , −9 , −8 , −7; 150 , 151 , 152 , 153")[$1 , 2 , 3 , 4 \  \ − 10 , −9 , −8 , −7 \ 150 , 151 , 152 , 153$]Notice that each number is one more than the number preceding it. So if we define the first integer as #emph[n], the next consecutive integer is #math.equation(block: false, alt: "n plus 1 .")[$n + 1 .$] The one after that is one more than #math.equation(block: false, alt: "n plus 1 ,")[$n + 1 ,$] so it is #math.equation(block: false, alt: "n plus 1 plus 1 ,")[$n + 1 + 1 ,$] which is #math.equation(block: false, alt: "n plus 2 .")[$n + 2 .$]

#math.equation(block: true, alt: "n, 1 to the power st integer; n plus 1, 2 to the power nd consecutive integer; n plus 2, 3 to the power rd consecutive integer . . . etc.")[$n &  &  & 1^("st") #h(0.2em) "integer" \ n + 1 &  &  & 2^("nd") #h(0.2em) "consecutive integer" \ n + 2 &  &  & 3^("rd") #h(0.2em) "consecutive integer . . . etc."$]#examplebox("Example 8")[][
The sum of two consecutive integers is 47. Find the numbers.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what you are looking for.], [], [two consecutive integers],
  [#emph[Step 3. Name] each number.], [], [Let #math.equation(block: false, alt: "n equals 1 to the power st")[$n = 1^("st")$] integer.],
  [], [], [#math.equation(block: false, alt: "n plus 1 equals")[$n + 1 =$] next consecutive integer],
  [#emph[Step 4. Translate.]], [], [],
  [Restate as one sentence.], [], [The sum of the integers is 47.],
  [Translate into an equation.], [], [#figure(figph[A basic algebraic equation is displayed, reading 'n + n + 1 = 47'. The equation is set against a plain white background, presenting a simple problem for solving for the variable 'n'.], alt: "A basic algebraic equation is displayed, reading 'n + n + 1 = 47'. The equation is set against a plain white background, presenting a simple problem for solving for the variable 'n'.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[The image displays a mathematical equation written in black text on a white background: n + n + 1 = 47.], alt: "The image displays a mathematical equation written in black text on a white background: n + n + 1 = 47.", caption: none)],
  [Combine like terms.], [], [#figure(figph[The image shows the mathematical equation 2n + 1 = 47, presented in a clear, sans-serif font against a plain white background.], alt: "The image shows the mathematical equation 2n + 1 = 47, presented in a clear, sans-serif font against a plain white background.", caption: none)],
  [Subtract 1 from each side.], [], [#figure(figph[A mathematical equation shows '2n = 46' in black text against a white background.], alt: "A mathematical equation shows '2n = 46' in black text against a white background.", caption: none)],
  [Divide each side by 2.], [], [#figure(figph[The image shows the mathematical expression 'n = 23 1st integer' on a white background.], alt: "The image shows the mathematical expression 'n = 23 1st integer' on a white background.", caption: none)],
  [], [], [#figure(figph[The text shows the mathematical expression 'n + 1' followed by the phrase 'next consecutive integer', indicating that n + 1 represents the integer that immediately follows n.], alt: "The text shows the mathematical expression 'n + 1' followed by the phrase 'next consecutive integer', indicating that n + 1 represents the integer that immediately follows n.", caption: none)],
  [], [], [#figure(figph[The numbers '23 + 1' are visible on a white background, representing a simple addition problem.], alt: "The numbers '23 + 1' are visible on a white background, representing a simple addition problem.", caption: none)],
  [], [], [#figure(figph[The number 24 is prominently displayed in black text on a clean white background.], alt: "The number 24 is prominently displayed in black text on a clean white background.", caption: none)],
  [#emph[Step 6. Check.]], [], [],
  [#math.equation(block: false, alt: "23 plus 24, ≟, 47; 47, equals, 47 ✓")[$23 + 24 & ≟ & 47 \ 47 & = & 47 ✓$]], [], [],
  [#emph[Step 7. Answer] the question.], [], [The two consecutive integers are 23 and 24.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two consecutive integers is #math.equation(block: false, alt: "95 .")[$95 .$] Find the numbers.

#solutionbox[
47, 48
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two consecutive integers is #math.equation(block: false, alt: "−31 .")[$−31 .$] Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−16 , −15")[$−16 , −15$]
]
]

#examplebox("Example 9")[][
Find three consecutive integers whose sum is #math.equation(block: false, alt: "−42 .")[$−42 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#emph[Step 1. Read] the problem.], [], [],
  [#emph[Step 2. Identify] what we are looking for.], [], [three consecutive integers],
  [#emph[Step 3. Name] each of the three numbers.], [], [Let #math.equation(block: false, alt: "n equals 1 to the power st")[$n = 1^("st")$] integer.],
  [], [], [#math.equation(block: false, alt: "n plus 1 equals")[$n + 1 =$] 2#super[nd] consecutive integer],
  [], [], [#math.equation(block: false, alt: "n plus 2 equals")[$n + 2 =$] 3#super[rd] consecutive integer],
  [#emph[Step 4. Translate.]], [], [],
  [Restate as one sentence.], [], [The sum of the three integers is −42.],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation is displayed on a white background: 'n + n + 1 + n + 2 = -42'. This equation simplifies to 3n + 3 = -42, which can be solved for n.], alt: "A mathematical equation is displayed on a white background: 'n + n + 1 + n + 2 = -42'. This equation simplifies to 3n + 3 = -42, which can be solved for n.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [#figure(figph[An algebraic equation is shown where n + n + 1 + n + 2 equals -42. This equation simplifies to 3n + 3 = -42, which further simplifies to 3n = -45, meaning n = -15.], alt: "An algebraic equation is shown where n + n + 1 + n + 2 equals -42. This equation simplifies to 3n + 3 = -42, which further simplifies to 3n = -45, meaning n = -15.", caption: none)],
  [Combine like terms.], [], [#figure(figph[A linear algebraic equation is shown, displaying '3n + 3 = -42' in black text on a white background, representing a problem to solve for the variable 'n'.], alt: "A linear algebraic equation is shown, displaying '3n + 3 = -42' in black text on a white background, representing a problem to solve for the variable 'n'.", caption: none)],
  [Subtract 3 from each side.], [], [#figure(figph[A mathematical equation is displayed, reading '3n = -45' against a white background.], alt: "A mathematical equation is displayed, reading '3n = -45' against a white background.", caption: none)],
  [Divide each side by 3.], [], [#figure(figph[The image shows the text 'n = -15 1st integer' on a white background.], alt: "The image shows the text 'n = -15 1st integer' on a white background.", caption: none)],
  [], [], [#linebreak() #figure(figph[The image displays the mathematical expression 'n+1' followed by the text '2nd integer' on a white background, suggesting a label or definition related to integer sequences or properties.], alt: "The image displays the mathematical expression 'n+1' followed by the text '2nd integer' on a white background, suggesting a label or definition related to integer sequences or properties.", caption: none)],
  [], [], [#figure(figph[The mathematical expression -15 + 1 is displayed on a white background, with '-15' in red and '+1' in black, suggesting a simple arithmetic problem.], alt: "The mathematical expression -15 + 1 is displayed on a white background, with '-15' in red and '+1' in black, suggesting a simple arithmetic problem.", caption: none)],
  [], [], [#figure(figph[The image displays the fraction '-1/4' in black text on a plain white background, appearing as if written or printed directly on the surface.], alt: "The image displays the fraction '-1/4' in black text on a plain white background, appearing as if written or printed directly on the surface.", caption: none)],
  [], [], [#linebreak() #figure(figph[The text on a white background says 'n+2 3rd integer'.], alt: "The text on a white background says 'n+2 3rd integer'.", caption: none)],
  [], [], [#figure(figph[The image displays a simple arithmetic problem, showing the expression '-15 + 2' in a clear, digital font against a white background.], alt: "The image displays a simple arithmetic problem, showing the expression '-15 + 2' in a clear, digital font against a white background.", caption: none)],
  [], [], [#figure(figph[The number -13 is prominently displayed in a neutral gray font against a plain white background.], alt: "The number -13 is prominently displayed in a neutral gray font against a plain white background.", caption: none)],
  [#emph[Step 6. Check.]], [], [],
  [#math.equation(block: false, alt: "−13 plus open parenthesis −14 close parenthesis plus open parenthesis −15 close parenthesis, ≟, −42; −42, equals, −42 ✓")[$−13 + ( −14 ) + ( −15 ) & ≟ & −42 \ −42 & = & −42 ✓$]], [], [],
  [#emph[Step 7. Answer] the question.], [], [The three consecutive integers are −13, −14, and −15.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three consecutive integers whose sum is #math.equation(block: false, alt: "−96 .")[$−96 .$]

#solutionbox[
#math.equation(block: true, alt: "−33 , −32 , −31")[$−33 , −32 , −31$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three consecutive integers whose sum is #math.equation(block: false, alt: "−36 .")[$−36 .$]

#solutionbox[
#math.equation(block: true, alt: "−13 , −12 , −11")[$−13 , −12 , −11$]
]
]

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. #emph[Consecutive even integers] are even integers that immediately follow one another. Examples of consecutive even integers are:

#math.equation(block: true, alt: "18 , 20 , 22; 64 , 66 , 68; −12 , −10 , −8")[$18 , 20 , 22 \  \ 64 , 66 , 68 \ −12 , −10 , −8$]Notice each integer is 2 more than the number preceding it. If we call the first one #emph[n], then the next one is #math.equation(block: false, alt: "n plus 2 .")[$n + 2 .$] The next one would be #math.equation(block: false, alt: "n plus 2 plus 2")[$n + 2 + 2$] or #math.equation(block: false, alt: "n plus 4 .")[$n + 4 .$]

#math.equation(block: true, alt: "n, 1 to the power st even integer; n plus 2, 2 to the power nd consecutive even integer; n plus 4, 3 to the power rd consecutive even integer . . . etc.")[$n &  &  & 1^("st") #h(0.2em) "even integer" \ n + 2 &  &  & 2^("nd") #h(0.2em) "consecutive even integer" \ n + 4 &  &  & 3^("rd") #h(0.2em) "consecutive even integer . . . etc."$]#emph[Consecutive odd integers] are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.

#math.equation(block: true, alt: "77 , 79 , 81; n , n plus 2 , n plus 4")[$77 , 79 , 81 \  \ n , n + 2 , n + 4$]#math.equation(block: true, alt: "n, 1 to the power st odd integer; n plus 2, 2 to the power nd consecutive odd integer; n plus 4, 3 to the power rd consecutive odd integer . . . etc.")[$n &  &  & 1^("st") #h(0.2em) "odd integer" \ n + 2 &  &  & 2^("nd") #h(0.2em) "consecutive odd integer" \ n + 4 &  &  & 3^("rd") #h(0.2em) "consecutive odd integer . . . etc."$]Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.

#examplebox("Example 10")[][
Find three consecutive even integers whose sum is 84.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what we are looking for.], [three consecutive even integers],
  [#emph[Step 3. Name] the integers.], [Let #math.equation(block: false, alt: "n equals 1 to the power st")[$n = 1^("st")$] even integer.
     #linebreak() #math.equation(block: false, alt: "n plus 2 equals 2 to the power nd")[$n + 2 = 2^("nd")$] consecutive even integer
     #linebreak() #math.equation(block: false, alt: "n plus 4 equals 3 to the power rd")[$n + 4 = 3^("rd")$] consecutive even integer],
  [#emph[Step 4. Translate.]], [],
  [Restate as one sentence.], [The sume of the three even integers is 84.],
  [Translate into an equation.], [#math.equation(block: false, alt: "n plus n plus 2 plus n plus 4 equals 84")[$n + n + 2 + n + 4 #h(0.2em) = #h(0.2em) 84$]],
  [#emph[Step 5. Solve] the equation.], [],
  [Combine like terms.], [#math.equation(block: false, alt: "n plus n plus 2 plus n plus 4 equals 84")[$n + n + 2 + n + 4 #h(0.2em) = #h(0.2em) 84$]],
  [Subtract 6 from each side.], [#math.equation(block: false, alt: "3 n plus 6 equals 84")[$3 n + 6 #h(0.2em) = #h(0.2em) 84$]],
  [Divide each side by 3.], [#math.equation(block: false, alt: "3 n equals 78")[$3 n #h(0.2em) = #h(0.2em) 78$]],
  [], [#math.equation(block: false, alt: "n equals 26, 1 to the power st integer; n plus 2, 2 to the power nd integer; 26 plus 2; 28; n plus 4, 3 to the power rd integer; 26 plus 4; 30")[$n = 26 & 1^("st") #h(0.2em) "integer" \  \ n + 2 & 2^("nd") #h(0.2em) "integer" \ 26 + 2 &  \ 28 &  \  \ n + 4 & 3^("rd") #h(0.2em) "integer" \ 26 + 4 &  \ 30 &$]],
  [#emph[Step 6. Check.]], [],
  [#math.equation(block: false, alt: "26 plus 28 plus 30 equals ? 84; 84 equals 84 ✓")[$26 + 28 + 30 limits(=)^(?) 84 \ 84 = 84 ✓$]], [],
  [#emph[Step 7. Answer] the question.], [The three consecutive integers are 26, 28, and 30.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three consecutive even integers whose sum is 102.

#solutionbox[
32, 34, 36
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three consecutive even integers whose sum is #math.equation(block: false, alt: "−24 .")[$−24 .$]

#solutionbox[
#math.equation(block: true, alt: "−10 , −8 , −6")[$−10 , −8 , −6$]
]
]

#examplebox("Example 11")[][
A married couple together earns \$110,000 a year. The wife earns \$16,000 less than twice what her husband earns. What does the husband earn?

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], [], []),
  [#emph[Step 2. Identify] what we are looking for.], [], [How much does the husband earn?],
  [#emph[Step 3. Name].], [], [],
  [Choose a variable to represent the amount #linebreak() the husband earns.], [], [Let #math.equation(block: false, alt: "h equals")[$h =$] the amount the husband earns.],
  [The wife earns \$16,000 less than twice that.], [], [#math.equation(block: false, alt: "2 h minus 16,000")[$2 h − 16,000$] the amount the wife earns.],
  [#emph[Step 4. Translate.]], [], [Together the husband and wife earn \$110,000.],
  [Restate the problem in one sentence with #linebreak() all the important information.], [], [#figure(figph[A visual representation of a couple's combined income, stating 'The amount the husband earns plus the amount the wife earns is \$110,000.'], alt: "A visual representation of a couple's combined income, stating 'The amount the husband earns plus the amount the wife earns is $110,000.'", caption: none)],
  [Translate into an equation.], [], [#figure(figph[A mathematical equation is displayed horizontally on a white background: h + 2h - 16,000 = 110,000.], alt: "A mathematical equation is displayed horizontally on a white background: h + 2h - 16,000 = 110,000.", caption: none)],
  [#emph[Step 5. Solve] the equation.], [], [h + 2h − 16,000 = 110,000],
  [Combine like terms.], [], [$#h(1.1em)$3h − 16,000 = 110,000],
  [Add 16,000 to both sides and simplify.], [], [$#h(4.7em)$3h = 126,000],
  [Divide each side by 3.], [], [$#h(5.2em)$h = 42,000],
  [], [], [$#h(6.2em)$\$42,000 amount husband earns],
  [], [], [$#h(4.7em)$2h − 16,000 amount wife earns],
  [], [], [$#h(1.88em)$2(42,000) − 16,000],
  [], [], [$#h(3.05em)$84,000 − 16,000],
  [], [], [$#h(6.85em)$68,000],
  [#emph[Step 6. Check.]], [], [],
  [If the wife earns \$68,000 and the husband earns \$42,000 is the total \$110,000? Yes!], [], [],
  [#emph[Step 7. Answer] the question.], [], [The husband earns \$42,000 a year.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
According to the National Automobile Dealers Association, the average cost of a car in 2014 was \$28,500. This was \$1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?

#solutionbox[
\$5,000
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
U.S. Census data shows that the median price of new home in the United States in November 2014 was \$280,900. This was \$10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

#solutionbox[
\$19,300
]
]


=== Key Concepts

- #strong[Problem-Solving Strategy]
+ #strong[Read] the problem. Make sure all the words and ideas are understood.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for. Choose a variable to represent that quantity.
+ #strong[Translate] into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.
- #strong[Consecutive Integers] #linebreak() 
Consecutive integers are integers that immediately follow each other. #linebreak() 
#math.equation(block: true, alt: "n, 1 to the power st integer; n plus 1, 2 to the power nd integer consecutive integer; n plus 2, 3 to the power rd consecutive integer . . . etc.")[$n &  &  & 1^("st") #h(0.2em) "integer" \ n + 1 &  &  & 2^("nd") #h(0.2em) "integer consecutive integer" \ n + 2 &  &  & 3^("rd") #h(0.2em) "consecutive integer . . . etc."$] #linebreak() 
Consecutive even integers are even integers that immediately follow one another. #linebreak() 
#math.equation(block: true, alt: "n, 1 to the power st integer; n plus 2, 2 to the power nd integer consecutive integer; n plus 4, 3 to the power rd consecutive integer . . . etc.")[$n &  &  & 1^("st") #h(0.2em) "integer" \ n + 2 &  &  & 2^("nd") #h(0.2em) "integer consecutive integer" \ n + 4 &  &  & 3^("rd") #h(0.2em) "consecutive integer . . . etc."$] #linebreak() 
Consecutive odd integers are odd integers that immediately follow one another. #linebreak() 
#math.equation(block: true, alt: "n, 1 to the power st integer; n plus 2, 2 to the power nd integer consecutive integer; n plus 4, 3 to the power rd consecutive integer . . . etc.")[$n &  &  & 1^("st") #h(0.2em) "integer" \ n + 2 &  &  & 2^("nd") #h(0.2em) "integer consecutive integer" \ n + 4 &  &  & 3^("rd") #h(0.2em) "consecutive integer . . . etc."$]


==== Practice Makes Perfect

#strong[Use the Approach Word Problems with a Positive Attitude]

In the following exercises, prepare the lists described.

List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

#solutionbox[
Answers will vary
]

List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

#strong[Use a Problem-Solving Strategy for Word Problems]

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?

#solutionbox[
30
]

Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?

Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

#solutionbox[
125
]

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

#solutionbox[
6
]

There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

#solutionbox[
58
]

Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?

Philip pays \$1,620 in rent every month. This amount is \$120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

#solutionbox[
\$750
]

Marc just bought an SUV for \$54,000. This is \$7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?

Laurie has \$46,000 invested in stocks and bonds. The amount invested in stocks is \$8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

#solutionbox[
\$13,500
]

Erica earned a total of \$50,450 last year from her two jobs. The amount she earned from her job at the store was \$1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

#strong[Solve Number Problems]

In the following exercises, solve each number word problem.

The sum of a number and eight is 12. Find the number.

#solutionbox[
4
]

The sum of a number and nine is 17. Find the number.

The difference of a number and 12 is three. Find the number.

#solutionbox[
15
]

The difference of a number and eight is four. Find the number.

The sum of three times a number and eight is 23. Find the number.

#solutionbox[
5
]

The sum of twice a number and six is 14. Find the number.

The difference of twice a number and seven is 17. Find the number.

#solutionbox[
12
]

The difference of four times a number and seven is 21. Find the number.

Three times the sum of a number and nine is 12. Find the number.

#solutionbox[
#math.equation(block: true, alt: "−5")[$−5$]
]

Six times the sum of a number and eight is 30. Find the number.

One number is six more than the other. Their sum is 42. Find the numbers.

#solutionbox[
18, 24
]

One number is five more than the other. Their sum is 33. Find the numbers.

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

#solutionbox[
8, 12
]

The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

The sum of two numbers is #math.equation(block: false, alt: "−45 .")[$−45 .$] One number is nine more than the other. Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−18 , −27")[$−18 , −27$]
]

The sum of two numbers is #math.equation(block: false, alt: "−61 .")[$−61 .$] One number is 35 more than the other. Find the numbers.

The sum of two numbers is #math.equation(block: false, alt: "−316 .")[$−316 .$] One number is 94 less than the other. Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−111 , −205")[$−111 , −205$]
]

The sum of two numbers is #math.equation(block: false, alt: "−284 .")[$−284 .$] One number is 62 less than the other. Find the numbers.

One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

#solutionbox[
32, 46
]

One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.

One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.

#solutionbox[
23, 28
]

One number is eight more than another. If their sum is increased by 17, the result is 95. Find the numbers.

One number is one more than twice another. Their sum is #math.equation(block: false, alt: "−5 .")[$−5 .$] Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−2 , −3")[$−2 , −3$]
]

One number is six more than five times another. Their sum is six. Find the numbers.

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

#solutionbox[
4, 10
]

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

The sum of two consecutive integers is 77. Find the integers.

#solutionbox[
38, 39
]

The sum of two consecutive integers is 89. Find the integers.

The sum of two consecutive integers is #math.equation(block: false, alt: "−23 .")[$−23 .$] Find the integers.

#solutionbox[
#math.equation(block: true, alt: "−11 , −12")[$−11 , −12$]
]

The sum of two consecutive integers is #math.equation(block: false, alt: "−37 .")[$−37 .$] Find the integers.

The sum of three consecutive integers is 78. Find the integers.

#solutionbox[
25, 26, 27
]

The sum of three consecutive integers is 60. Find the integers.

Find three consecutive integers whose sum is #math.equation(block: false, alt: "−36 .")[$−36 .$]

#solutionbox[
#math.equation(block: true, alt: "−11 , −12 , −13")[$−11 , −12 , −13$]
]

Find three consecutive integers whose sum is #math.equation(block: false, alt: "−3 .")[$−3 .$]

Find three consecutive even integers whose sum is 258.

#solutionbox[
84, 86, 88
]

Find three consecutive even integers whose sum is 222.

Find three consecutive odd integers whose sum is 171.

#solutionbox[
55, 57, 59
]

Find three consecutive odd integers whose sum is 291.

Find three consecutive even integers whose sum is #math.equation(block: false, alt: "−36 .")[$−36 .$]

#solutionbox[
#math.equation(block: true, alt: "−10 , −12 , −14")[$−10 , −12 , −14$]
]

Find three consecutive even integers whose sum is #math.equation(block: false, alt: "−84 .")[$−84 .$]

Find three consecutive odd integers whose sum is #math.equation(block: false, alt: "−213 .")[$−213 .$]

#solutionbox[
#math.equation(block: true, alt: "−69 , −71 , −73")[$−69 , −71 , −73$]
]

Find three consecutive odd integers whose sum is #math.equation(block: false, alt: "−267 .")[$−267 .$]


==== Everyday Math

#strong[Sale Price] Patty paid \$35 for a purse on sale for \$10 off the original price. What was the original price of the purse?

#solutionbox[
\$45
]

#strong[Sale Price] Travis bought a pair of boots on sale for \$25 off the original price. He paid \$60 for the boots. What was the original price of the boots?

#strong[Buying in Bulk] Minh spent \$6.25 on five sticker books to give his nephews. Find the cost of each sticker book.

#solutionbox[
\$1.25
]

#strong[Buying in Bulk] Alicia bought a package of eight peaches for \$3.20. Find the cost of each peach.

#strong[Price before Sales Tax] Tom paid \$1,166.40 for a new refrigerator, including \$86.40 tax. What was the price of the refrigerator?

#solutionbox[
\$1080
]

#strong[Price before Sales Tax] Kenji paid \$2,279 for a new living room set, including \$129 tax. What was the price of the living room set?


==== Writing Exercises

What has been your past experience solving word problems?

#solutionbox[
answers will vary
]

When you start to solve a word problem, how do you decide what to let the variable represent?

What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.

#solutionbox[
Consecutive odd integers are odd numbers that immediately follow each other. An example of three consecutive odd integers between 50 and 60 would be 51, 53, and 55.
]

What are consecutive even integers? Name three consecutive even integers between #math.equation(block: false, alt: "−50")[$−50$] and #math.equation(block: false, alt: "−40 .")[$−40 .$]


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “approach word problems with a positive attitude,” use a problem solving strategy for word problems,” and “solve number problems.” The rest of the cells are blank], alt: "This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “approach word problems with a positive attitude,” use a problem solving strategy for word problems,” and “solve number problems.” The rest of the cells are blank", caption: none)

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.